1. Key Stage 3 National Strategy Interacting with mathematics in Key Stage 3

2. Key Stage 3 National Strategy Objectives for the course To explain the purpose and use of the Year 9 materials To explain the purpose and use of the Year 9 materials To explore the mini-packs on geometrical reasoning and proportional reasoning To explore the mini-packs on geometrical reasoning and proportional reasoning To consider effective ways of using the materials To consider effective ways of using the materials Slide CT1

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

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

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

6. Key Stage 3 National Strategy Geometrical reasoning: objectives for meeting 1 To indicate some current issues in teaching geometry To indicate some current issues in teaching geometry To try a visualisation and reconstruction exercise To try a visualisation and reconstruction exercise To outline a Year 9 unit on geometrical reasoning To outline a Year 9 unit on geometrical reasoning To consider how to help pupils work towards proof in geometry To consider how to help pupils work towards proof in geometry Slide GR1

7. Key Stage 3 National Strategy Key features of the geometrical reasoning unit Provides six hours (about two weeks) of work Provides six hours (about two weeks) of work Aims to draw together pupils geometrical knowledge into a logical and reasoned framework Aims to draw together pupils geometrical knowledge into a logical and reasoned framework Develops pupils visualisation skills through oral and mental starters Develops pupils visualisation skills through oral and mental starters Slide GR2

8. Key Stage 3 National Strategy Key features of the geometrical reasoning unit Phase 1 uses build-ups to develop pupils ability to reason logically and prove results Phase 1 uses build-ups to develop pupils ability to reason logically and prove results Phase 2 focuses on solving problems Phase 2 focuses on solving problems Suggests activities for each phase Suggests activities for each phase Slide GR3

9. Key Stage 3 National Strategy Using build-ups: towards proof Given facts are used to build up new geometrical properties Given facts are used to build up new geometrical properties All use conventions All use conventions All give the opportunity to rehearse both oral and written forms of an argument All give the opportunity to rehearse both oral and written forms of an argument Some establish definitions Some establish definitions Some prove properties from statements of given facts established previously Some prove properties from statements of given facts established previously Slide GR4

10. Key Stage 3 National Strategy Using build-ups: towards proof Pupils need to appreciate: the nature of the game the nature of the game that they will be asked to justify results that they will be asked to justify results the importance of having a logical order to their argument the importance of having a logical order to their argument Slide GR5

11. Key Stage 3 National Strategy Pupils discussing build-ups As you listen to the recording: reconstruct conventions used by pupils to label the diagram reconstruct conventions used by pupils to label the diagram identify areas of vocabulary which could be more precise identify areas of vocabulary which could be more precise be prepared to record a written version of the stages of their proof be prepared to record a written version of the stages of their proof Slide GR6

12. Key Stage 3 National Strategy Geometrical reasoning: objectives for meeting 2 To revisit the use of visualisation exercises To revisit the use of visualisation exercises To explore approaches to solving geometrical problems To explore approaches to solving geometrical problems To decide on practical steps needed to implement the unit To decide on practical steps needed to implement the unit To discuss any general implications for the work of the department To discuss any general implications for the work of the department Slide GR7

13. Key Stage 3 National Strategy Slide GR8 Pupils discussing visualisations After the recording, discuss: how an open discussion can reveal areas for further work, such as developing language or resolving misconceptions 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 the value of working with dynamic images in the mind and sharing perceptions with others

14. Key Stage 3 National Strategy 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 structure the stages of solving a geometrical problem provide strategies to support the development of written argument provide strategies to support the development of written argument Slide GR9

15. Key Stage 3 National Strategy Solving geometrical problems The problem bank: contains questions of various types, including alternative wording of particular questions contains questions of various types, including alternative wording of particular questions is not intended to be copied directly as a worksheet is not intended to be copied directly as a worksheet is available on the Year 9 CD-ROM as a text file is available on the Year 9 CD-ROM as a text file Slide GR10

16. Key Stage 3 National Strategy Reflecting on the geometrical reasoning 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 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? 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. Key Stage 3 National Strategy Proportional reasoning: objectives for meeting 1 To review the effectiveness of the Year 8 unit on multiplicative relationships To review the effectiveness of the Year 8 unit on multiplicative relationships To explore the Year 9 unit on proportional reasoning To explore the Year 9 unit on proportional reasoning To discuss teaching of the main activities in phases 1 and 2 of the unit To discuss teaching of the main activities in phases 1 and 2 of the unit Slide PR1

18. Key Stage 3 National Strategy Key features of the proportional reasoning unit Provides nine hours (about three weeks) of work Provides nine hours (about three weeks) of work Links closely with the multiplicative relationships unit Links closely with the multiplicative relationships unit Slide PR2

19. Key Stage 3 National Strategy Key features of the proportional reasoning unit Phase 1 revisits the idea of scaling numbers and using tabulation to solve proportion problems 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 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 Phase 3 consolidates and extends strategies for solving problems Slide PR3

20. Key Stage 3 National Strategy Key features of the proportional reasoning unit Oral and mental starters work towards automating key aspects of calculating with proportions Oral and mental starters work towards automating key aspects of calculating with proportions Activities are suggested for each phase Activities are suggested for each phase Slide PR4

21. Key Stage 3 National Strategy Scaling and proportional sets Phase 1 builds on the Year 8 work on multiplicative relationships Phase 1 builds on the Year 8 work on multiplicative relationships The idea of scaling numbers is extended to repeated scalings The idea of scaling numbers is extended to repeated scalings Tables of numbers in proportion are revisited and extended Tables of numbers in proportion are revisited and extended Slide PR5

22. Key Stage 3 National Strategy Enlargement and similarity Enlargement and similarity are included because: links between different aspects of mathematics can be made explicitly links between different aspects of mathematics can be made explicitly both provide visual images to extend pupils understanding of proportion both provide visual images to extend pupils understanding of proportion Slide PR6

23. Key Stage 3 National Strategy Enlargement and similarity Prompts for three main activities in phase 2: Folding paper Folding paper Cat faces Cat faces Photographic enlargements Photographic enlargements Prompts for the final plenary in phase 2: Shadows Shadows Slide PR7

24. Key Stage 3 National Strategy Proportional reasoning: objectives for meeting 2 To consider the purpose of oral and mental starters in this unit To consider the purpose of oral and mental starters in this unit To discuss how to teach the techniques required to solve proportion problems To discuss how to teach the techniques required to solve proportion problems To decide on practical steps needed to implement the unit To decide on practical steps needed to implement the unit To discuss any general implications for the work of the department To discuss any general implications for the work of the department Slide PR8

25. Key Stage 3 National Strategy Key features of the oral and mental starters Activities should be used selectively Activities should be used selectively Activities rehearse skills or ideas which are precursors to later work Activities rehearse skills or ideas which are precursors to later work Some processes need to be automated Some processes need to be automated Developing fluency requires pace Developing fluency requires pace Slide PR9

26. Key Stage 3 National Strategy 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 rates and change of units problems involving percentages problems involving percentages using and applying proportional relationships in different contexts using and applying proportional relationships in different contexts Slide PR10

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

28. Key Stage 3 National Strategy Reflecting on the proportional reasoning unit What effective teaching strategies used by Raj could you consider adopting in your department? 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 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? How has this unit suggested ways of making further links between strands of the yearly teaching programmes? Slide PR12