Key Stage 3 National Strategy

1. Key Stage 3 National Strategy Interacting with mathematicsin Key Stage 3

2. Objectives for the course • To explain the purpose and use of the Year 9 materials • To explore the mini-packs on geometrical reasoning and proportional reasoning • To consider effective ways of using the materials Slide CT1

3. Interacting with mathematicsin Year 9: aims To support mathematics departments in: • planning for teaching that engages and challenges pupils • developing mathematical reasoning Slide CT2

4. Year 8 Handling data Multiplicative relationships Year 9 Geometrical reasoning Proportional reasoning Interacting with mathematicsin Key Stage 3: materials Slide CT3

5. Materials available on the web Address: www.standards.dfes.gov.uk/keystage3 Choose: • mathematics • mathematics publications • Interacting with mathematics in Key Stage 3 Slide CT4

6. Geometrical reasoning: objectives for meeting 1 • To indicate some current issues in teaching geometry • To try a visualisation and reconstruction exercise • To outline a Year 9 unit on geometrical reasoning • To consider how to help pupils work towards proof in geometry Slide GR1

7. Key features of the geometricalreasoning unit • Provides six hours (about two weeks) of work • Aims to draw together pupils’ geometrical knowledge into a logical and reasoned framework • Develops pupils’ visualisation skills through oral and mental starters Slide GR2

8. Key features of the geometricalreasoning unit • Phase 1 uses ‘build-ups’ to develop pupils’ ability to reason logically and prove results • Phase 2 focuses on solving problems • Suggests activities for each phase Slide GR3

9. Using ‘build-ups’: towards proof • Given facts are used to ‘build up’ new geometrical properties • All use conventions • All give the opportunity to rehearse both oral and written forms of an argument • Some establish definitions • Some prove properties from statements of given facts established previously Slide GR4

10. Using ‘build-ups’: towards proof Pupils need to appreciate: • the nature of the ‘game’ • that they will be asked to justify results • the importance of having a logical order to their argument Slide GR5

11. Pupils discussing build-ups As you listen to the recording: • reconstruct conventions used by pupils to label the diagram • identify areas of vocabulary which could be more precise • be prepared to record a written version of the stages of their proof Slide GR6

12. Geometrical reasoning: objectives for meeting 2 • To revisit the use of visualisation exercises • To explore approaches to solving geometrical problems • To decide on practical steps needed to implement the unit • To discuss any general implications for the work of the department Slide GR7

13. Pupils discussing visualisations After the recording, discuss: • how an open discussion can reveal areas for further work, such as developing language or resolving misconceptions • the value of working with dynamic images in the mind and sharing perceptions with others Slide GR8

14. Solving geometrical problems ‘Prompts for main activities in phase 2: solving problems’ offers guidance on how to: • structure the stages of solving a geometrical problem • provide strategies to support the development of written argument Slide GR9

15. Solving geometrical problems The problem bank: • contains questions of various types, including alternative wording of particular questions • is not intended to be copied directly as a worksheet • is available on the Year 9 CD-ROM as a text file Slide GR10

16. Reflecting on the geometricalreasoning unit • Has the link between visual images, layers of a built-up diagram and structuring of oral and written argument helped to clarify the process of geometrical reasoning? • Has the content of this unit suggested approaches to teaching geometrical reasoning that might be extended into Key Stage 4 and filtered down into Year 8? Slide GR11

17. Proportional reasoning: objectives for meeting 1 • To review the effectiveness of the Year 8 unit on multiplicative relationships • To explore the Year 9 unit on proportional reasoning • To discuss teaching of the main activities in phases 1and 2 of the unit Slide PR1

18. Key features of the proportionalreasoning unit • Provides nine hours (about three weeks) of work • Links closely with the multiplicative relationships unit Slide PR2

19. Key features of the proportionalreasoning unit • Phase 1 revisits the idea of scaling numbers and using tabulation to solve proportion problems • Phase 2 offers images for proportion and scaling by making links with enlargement and similarity in geometry • Phase 3 consolidates and extends strategies for solving problems Slide PR3

20. Key features of the proportionalreasoning unit • Oral and mental starters work towards automating key aspects of calculating with proportions • Activities are suggested for each phase Slide PR4

21. Scaling and proportional sets • Phase 1 builds on the Year 8 work on multiplicative relationships • The idea of scaling numbers is extended to repeated scalings • Tables of numbers in proportion are revisited and extended Slide PR5

22. Enlargement and similarity Enlargement and similarity are included because: • links between different aspects of mathematics can be made explicitly • both provide visual images to extend pupils’ understanding of proportion Slide PR6

23. Enlargement and similarity Prompts for three main activities in phase 2: • Folding paper • Cat faces • Photographic enlargements Prompts for the final plenary in phase 2: • Shadows Slide PR7

24. Proportional reasoning: objectives for meeting 2 • To consider the purpose of oral and mental starters inthis unit • To discuss how to teach the techniques required to solve proportion problems • To decide on practical steps needed to implement the unit • To discuss any general implications for the work of the department Slide PR8

25. Key features of the oral and mental starters • Activities should be used selectively • Activities rehearse skills or ideas which are precursors to later work • Some processes need to be automated • Developing fluency requires pace Slide PR9

26. Teaching problem solving Year 9 materials build on problem-solving approaches introduced in Year 8 Year 9 mini-pack includes additional suggestions for: • problems involving rates and change of units • problems involving percentages • using and applying proportional relationships in different contexts Slide PR10

27. The problem bank You will need to collect a bank of problems for phase 3. Suggested sources include: • Year 8 problem bank • Framework supplement of examples • questions from Key Stage 3 tests • textbooks Slide PR11

28. Reflecting on the proportionalreasoning unit • What effective teaching strategies used by Raj could you consider adopting in your department? • How has the structure of this unit and the Year 8 unit suggested a model for tracking through a mathematical topic across two or more years? • How has this unit suggested ways of making further links between strands of the yearly teaching programmes? Slide PR12