Mathematics guidance: key stage 3

1 Mathematics guidance: Key stage 3 Non-statutory guidance for the national curriculum in England September 2021 2 Acknowledgements We would like to thank and acknowledge the following people involved in the production of this publication: Debbie Barker; Frances Carr; Alf Coles; Clare Dawson; Becky Donaldson; Pete Griffin; Jane Hawkins; Alison Hopper; Rachel Houghton; Carol Knights; Steve Lomax; Steve McCormack; Richard Perring; Pete Sides; Dr Mary Stevenson; Charlie Stripp; Dr Nicola Trubridge; Andrew Young and the Maths Hubs Secondary Mastery Specialists. All illustrations by Steve Evans NCETM unless otherwise stated. Extracts from Mathematics past papers from Standards & Testing Agency and other Public sector information licensed under the Open Government Licence Extracts from the national curriculum: Department for Education, 2013, National curriculum in England: Mathematics programmes of study Graph of world population from Reference made to the work of: E.

2 Gray & D. Tall, (1991) Duality, Ambiguity and Flexibility in Successful mathematical Thinking, (Coventry, University of Warwick); K. M. Hart (ed.), (1981) Children's Understanding of Mathematics : 11 16, (London, John Murray); K chemann, D. (1978) Children s Understanding of Numerical Variables, Mathematics in School, 7(4), 23 26; T. Nunes & P. Bryant, (2009) Paper 3: Understanding rational numbers and intensive quantities. In T. Nunes, P. Bryant and A. Watson, Key understandings in Mathematics learning: A report to the Nuffield Foundation, ; Mitchelmore & P. White, (2000) Development of angle concepts by progressive abstraction and generalisation, Educational Studies in Mathematics , 41, 209 38; D. Pratt & R. Noss, (2002), The Microevolution of mathematical Knowledge: The Case of Randomness, Journal of the Learning Sciences, 11(4), 453 488, (Philadelphia, Taylor & Francis). 3 Contents Acknowledgements 2 Contents 3 Summary 5 Who is this publication for?

3 5 Aims 6 Structure of the document 7 Designing a coherent and connected curriculum 10 Purpose and rationale 10 Guidance 11 Sample Key stage 3 curriculum framework 16 Year 7 sample curriculum framework 19 Year 8 sample curriculum framework 21 Year 9 sample curriculum framework 23 Split statements of knowledge, skills and understanding 24 Year 7 autumn term 26 Place value 26 Properties of number: factors, multiples, squares and cubes 30 Arithmetic procedures with integers and decimals 41 Expressions and equations 56 Year 7 spring term 67 Plotting coordinates 67 Perimeter and area 73 Arithmetic procedures including fractions 80 Year 7 summer term 98 Understanding multiplicative relationships: fractions and ratio 98 Transformations 112 Year 8 autumn term 121 Estimation and rounding 121 Sequences 129 Graphical representations of linear relationships 137 4 Solving linear equations 147 Year 8 spring term 160 Understanding multiplicative relationships: percentages and proportionality 160 Statistical representations and measures 167 Statistical analysis 175 Year 8 summer term 183 Perimeter, area and volume 183 Geometrical properties: polygons 195 Constructions 204 Year 9 autumn term 212 Geometrical properties: similarity and Pythagoras theorem 212 Probability 222 Year 9 spring term 230 Non-linear relationships 230 Expressions and formulae 233 Trigonometry 241 Year 9 summer term 253 Standard form 253 Graphical representations 258 Appendix 1 key ideas 267 Appendix 2 language 279 5 Summary This publication provides non-statutory guidance from the Department for Education.

4 It has been produced to help teachers and schools make effective use of the national curriculum to develop secondary school pupils mastery of Mathematics . Who is this publication for? This guidance is for: local authorities school leaders, school staff and governing bodies in all maintained schools, academies and free schools. 6 Aims This publication aims to: Bring greater coherence to the national curriculum for Mathematics by exemplifying the statutory guidance for Key stage 3 (DfE, 2013) and giving schools, Mathematics departments and teachers further guidance on how learning in Mathematics develops across Key stage 3. Highlight the most important knowledge and understanding developed during Key stage 3, the connections between different mathematical topics, and how they link back to Key stage 2 and forward to Key stage 4. Key considerations concerning how to distribute the national curriculum content across the key stage are discussed.

5 A sample model of a curriculum framework is provided to help Mathematics departments structure teaching and learning effectively. Guidance is given on how teachers within a school s Mathematics department might collaborate to plan their long- and medium-term teaching. Fundamental concepts are highlighted, including how they build on content learnt in Key stage 2; how they will be developed in Key stage 4; and which aspects should be prioritised and consolidated within Key stage 3. For selected key ideas, detailed guidance is provided, including common misconceptions; teaching approaches that lead to a deep and connected understanding; and sample questions. Teaching and learning are complex, but the intention should always be to develop students understanding of mathematical concepts and structures, alongside providing sufficient practice to attain fluency. This combination of developing fluency and mathematical understanding in tandem will enable students to use their learning accurately, efficiently and flexibly to reason mathematically and solve routine and non-routine problems, so meeting the aims of the national curriculum.

6 The guidance in this document and examples of practice provide teachers with a variety of ways to offer their students opportunities to develop their mathematical understanding and skills. Teachers should also supplement this learning activity with opportunities for students to apply their knowledge to questions where a method for solution is not immediately obvious, but draws upon previously mastered Mathematics . 7 Structure of the document The first sections of this document give guidance to departments about long-term planning for cohesive student learning. Later sections are structured termly by year group, according to the order of teaching suggested in the sample curriculum framework offered on pages 16 to 25 in this document. Each termly section is structured as follows: Overview This section outlines how the content fits with wider mathematical learning and identifies key considerations and emphases for teaching. Prior learning Year 7 learning is built upon the mathematical foundations established in Key stage 2, while some later study in Key stage 3 is also underpinned by mathematical concepts encountered earlier in the key stage .

7 The sample curriculum framework in this document is used as a basis for an order of teaching and the suggested prior learning elements align with this progression. Prior learning identified from Key stage 2 includes the ready-to-progress criteria outlined in the Key stage 2 non-statutory guidance. The ready-to-progress statements are divided into strands as follows: Ready-to-progress criteria strands Code Number and place value NPV Number facts NF Addition and subtraction AS Multiplication and division MD Fractions F Geometry G The code 6-AS refers to elements of the addition and subtraction strand that are recommended to be learnt in Year 6. Most of the ready-to-progress criteria referred to are 8 from Year 6, but some areas of mathematical content from Years 4 and 5 are referenced where it is unlikely that there has been further study of these elements in Year 6. Checking prior learning It is of paramount importance that teaching, at all stages, takes account of prior learning and the depth of understanding already achieved.

8 Pitching teaching appropriately ensures that students are neither bored by repeating content which is already well understood, nor flummoxed by content which they cannot readily assimilate with their existing knowledge. Achieving this balance is arguably one of the most challenging elements of teaching. Using formative assessment approaches to check prior learning before teaching a particular sequence of lessons can inform effective planning. In this section, sample questions are provided which teachers can use to probe the depth of prior attainment. Language Using correct mathematical language and terminology gives students the fundamental tools to communicate their reasoning, thinking and ideas accurately and precisely, avoiding ambiguity and potential confusion. Modelling and encouraging the use of correct mathematical language will support students in using it confidently. A selection of key words and phrases that should be encouraged within Key stage 3 is listed in each section and explained in Appendix 2.

9 Progression through key ideas In the sample curriculum framework there are between two and four core concepts to be taught in each school term. These are broken down into statements of knowledge, skills and understanding that students should aim to achieve. These statements are further divided into key ideas, which are listed in this section throughout. The broader breakdown will help with medium-term planning, whereas the more detailed breakdown will aid short-term planning. There is no suggestion that that each key idea represents a lesson. The amount of classroom time required for different key ideas to be mastered will vary. Developing a deep and connected understanding of these key ideas will enable students to make secure progress through the curriculum. Key ideas with an asterisk * after them are then exemplified. A full list of key ideas can be found in Appendix 1. 9 Exemplified significant key ideas For a selection of significant key ideas, further teaching guidance is given.

10 Within each exemplification, consideration is given to the common difficulties teachers may encounter and the misconceptions students may hold. Teaching approaches and useful representations are also considered. A few examples of questions which could be used within teaching are then given together with some commentary. The commentary may draw attention to: how the particular example might be used; how it is structured and why this is beneficial to learners; misconceptions that it will help elicit; how the representation used will aid students in deepening their understanding; or some other aspect of pedagogy. The examples included in this document are drawn from the NCETM s Key stage 3 Professional Development materials; further examples and commentary are offered within these materials. 10 Designing a coherent and connected curriculum Purpose and rationale High-quality teaching of Mathematics in the classroom is, of course, what really makes a difference to students learning.