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

Key Stage 3 National Strategy

Key Stage 3 Key Stage 3 National StrategyNational Strategy

Interacting with mathematicsInteracting with mathematicsin Key Stage 3in Key Stage 3


Objectives for the courseObjectives for the course

To explain the purpose and use of the Year 9 To explain the purpose and use of the Year 9 materialsmaterials

To explore the mini-packs on geometrical To explore the mini-packs on geometrical reasoning and proportional reasoningreasoning and proportional reasoning

To consider effective ways of using the To consider effective ways of using the materialsmaterials

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Interacting with mathematicsInteracting with mathematicsin Year 9: aimsin Year 9: aims

To support mathematics departments in:To support mathematics departments in:

planning for teaching that engages and planning for teaching that engages and challenges pupilschallenges pupils

developing mathematical reasoningdeveloping mathematical reasoning

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Interacting with mathematicsInteracting with mathematicsin Key Stage 3: materialsin Key Stage 3: materials

Year 8Year 8

Handling dataHandling data

Multiplicative relationshipsMultiplicative relationships

Year 9Year 9

Geometrical reasoningGeometrical reasoning

Proportional reasoningProportional reasoning

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Materials available on the webMaterials available on the web

Address: www.standards.dfes.gov.uk/keystage3Address: www.standards.dfes.gov.uk/keystage3

Choose:Choose:

mathematicsmathematics

mathematics publicationsmathematics publications

Interacting with mathematics in Key Stage 3Interacting with mathematics in Key Stage 3

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Geometrical reasoning: Geometrical reasoning: objectives for meeting 1objectives for meeting 1

To indicate some current issues in teaching To indicate some current issues in teaching geometrygeometry

To try a visualisation and reconstruction exerciseTo try a visualisation and reconstruction exercise

To outline a Year 9 unit on geometrical reasoningTo outline a Year 9 unit on geometrical reasoning

To consider how to help pupils work towards To consider how to help pupils work towards proof in geometryproof in geometry

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Key features of the geometricalKey features of the geometricalreasoning unitreasoning unit

Provides six hours (about two weeks) of Provides six hours (about two weeks) of workwork

Aims to draw together pupils’ geometrical Aims to draw together pupils’ geometrical knowledge into a logical and reasoned knowledge into a logical and reasoned frameworkframework

Develops pupils’ visualisation skills through Develops pupils’ visualisation skills through oral and mental startersoral and mental starters

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Key features of the geometricalKey features of the geometricalreasoning unitreasoning unit

Phase 1 uses ‘build-ups’ to develop Phase 1 uses ‘build-ups’ to develop pupils’ ability to reason logically and pupils’ ability to reason logically and prove resultsprove results

Phase 2 focuses on solving problemsPhase 2 focuses on solving problems

Suggests activities for each phaseSuggests activities for each phase

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Using ‘build-ups’: towards proofUsing ‘build-ups’: towards proof Given facts are used to ‘build up’ new geometrical Given facts are used to ‘build up’ new geometrical

propertiesproperties

All use All use conventionsconventions

All give the opportunity to rehearse both oral and All give the opportunity to rehearse both oral and written forms of an argumentwritten forms of an argument

Some establish Some establish definitionsdefinitions

Some prove Some prove properties properties from statements of given from statements of given facts established previouslyfacts established previously

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Using ‘build-ups’: towards proofUsing ‘build-ups’: towards proof

Pupils need to appreciate:Pupils need to appreciate:

the nature of the ‘game’the nature of the ‘game’

that they will be asked to justify resultsthat they will be asked to justify results

the importance of having a logical order to the importance of having a logical order to their argumenttheir argument

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Pupils discussing build-upsPupils discussing build-ups

As you listen to the recording:As you listen to the recording:

reconstruct conventions used by pupils to label reconstruct conventions used by pupils to label the diagramthe diagram

identify areas of vocabulary which could be identify areas of vocabulary which could be more precisemore precise

be prepared to record a written version of the be prepared to record a written version of the stages of their proofstages of their proof

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Geometrical reasoning: Geometrical reasoning: objectives for meeting 2objectives for meeting 2

To revisit the use of visualisation exercisesTo revisit the use of visualisation exercises

To explore approaches to solving geometrical To explore approaches to solving geometrical problemsproblems

To decide on practical steps needed to To decide on practical steps needed to implement the unitimplement the unit

To discuss any general implications for the work To discuss any general implications for the work of the departmentof the department

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Key Stage 3 National StrategySlide GR8

Pupils discussing visualisationsPupils discussing visualisations

After the recording, discuss:After the recording, discuss:

how an open discussion can reveal areas how an open discussion can reveal areas for further work, such as developing for further work, such as developing language or resolving misconceptionslanguage or resolving misconceptions

the value of working with dynamic images the value of working with dynamic images in the mind and sharing perceptions with in the mind and sharing perceptions with othersothers


Solving geometrical problemsSolving geometrical problems

‘‘Prompts for main activities in phase 2: Prompts for main activities in phase 2: solving problems’solving problems’

offers guidance on how to:offers guidance on how to:

structure the stages of solving a structure the stages of solving a geometrical problemgeometrical problem

provide strategies to support the provide strategies to support the development of written argumentdevelopment of written argument

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Solving geometrical problemsSolving geometrical problems

The problem bank:The problem bank:

contains questions of various types, including contains questions of various types, including alternative wording of particular questionsalternative wording of particular questions

is not intended to be copied directly as a is not intended to be copied directly as a worksheetworksheet

is available on the Year 9 CD-ROM as a text fileis available on the Year 9 CD-ROM as a text file

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Reflecting on the geometricalReflecting on the geometricalreasoning unitreasoning unit

Has the link between visual images, layers of a Has the link between visual images, layers of a built-up diagram and structuring of oral and built-up diagram and structuring of oral and written argument helped to clarify the process of written argument helped to clarify the process of geometrical reasoning?geometrical reasoning?

Has the content of this unit suggested Has the content of this unit suggested approaches to teaching geometrical reasoning approaches to teaching geometrical reasoning that might be extended into Key Stage 4 and that might be extended into Key Stage 4 and filtered down into Year 8?filtered down into Year 8?

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Proportional reasoning: Proportional reasoning: objectives for meeting 1objectives for meeting 1

To review the effectiveness of the Year 8 To review the effectiveness of the Year 8 unit on multiplicative relationshipsunit on multiplicative relationships

To explore the Year 9 unit on proportional To explore the Year 9 unit on proportional reasoningreasoning

To discuss teaching of the main activities in To discuss teaching of the main activities in phases 1phases 1and 2 of the unitand 2 of the unit

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Key features of the proportionalKey features of the proportionalreasoning unitreasoning unit

Provides nine hours (about three weeks) of Provides nine hours (about three weeks) of workwork

Links closely with the multiplicative Links closely with the multiplicative relationships unitrelationships unit

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Phase 1 revisits the idea of scaling numbers and Phase 1 revisits the idea of scaling numbers and using tabulation to solve proportion problemsusing tabulation to solve proportion problems

Phase 2 offers images for proportion and scaling Phase 2 offers images for proportion and scaling by making links with enlargement and similarity in by making links with enlargement and similarity in geometrygeometry

Phase 3 consolidates and extends strategies for Phase 3 consolidates and extends strategies for solving problemssolving problems

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Oral and mental starters work towards Oral and mental starters work towards automating key aspects of calculating with automating key aspects of calculating with proportionsproportions

Activities are suggested for each phaseActivities are suggested for each phase

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Scaling and proportional setsScaling and proportional sets

Phase 1 builds on the Year 8 work on Phase 1 builds on the Year 8 work on multiplicative relationshipsmultiplicative relationships

The idea of scaling numbers is extended to The idea of scaling numbers is extended to repeated scalingsrepeated scalings

Tables of numbers in proportion are Tables of numbers in proportion are revisited and extendedrevisited and extended

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Enlargement and similarityEnlargement and similarity

Enlargement and similarity are included Enlargement and similarity are included because:because:

links between different aspects of links between different aspects of mathematics can be made explicitlymathematics can be made explicitly

both provide visual images to extend both provide visual images to extend pupils’ understanding of proportionpupils’ understanding of proportion

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Enlargement and similarityEnlargement and similarity

Prompts for three main activities in phase 2:Prompts for three main activities in phase 2:

Folding paperFolding paper

Cat facesCat faces

Photographic enlargementsPhotographic enlargements

Prompts for the final plenary in phase 2:Prompts for the final plenary in phase 2:

ShadowsShadows

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Proportional reasoning: Proportional reasoning: objectives for meeting 2objectives for meeting 2 To consider the purpose of oral and mental starters inTo consider the purpose of oral and mental starters in

this unitthis unit

To discuss how to teach the techniques required to To discuss how to teach the techniques required to solve proportion problemssolve proportion problems

To decide on practical steps needed to implement the To decide on practical steps needed to implement the unitunit

To discuss any general implications for the work of the To discuss any general implications for the work of the departmentdepartment

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Key features of the oral and mental Key features of the oral and mental startersstarters

Activities should be used selectivelyActivities should be used selectively

Activities rehearse skills or ideas which are Activities rehearse skills or ideas which are precursors to later workprecursors to later work

Some processes need to be automatedSome processes need to be automated

Developing fluency requires paceDeveloping fluency requires pace

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Teaching problem solvingTeaching problem solving

Year 9 materials build on problem-solving approachesYear 9 materials build on problem-solving approachesintroduced in Year 8introduced in Year 8

Year 9 mini-pack includes additional suggestions for:Year 9 mini-pack includes additional suggestions for:

problems involving rates and change of unitsproblems involving rates and change of units

problems involving percentagesproblems involving percentages

using and applying proportional relationships in using and applying proportional relationships in different contextsdifferent contexts

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The problem bankThe problem bank

You will need to collect a bank of problems for phase You will need to collect a bank of problems for phase 3.3.

Suggested sources include: Suggested sources include:

Year 8 problem bankYear 8 problem bank

Framework supplement of examplesFramework supplement of examples

questions from Key Stage 3 testsquestions from Key Stage 3 tests

textbookstextbooksSlide PR11


Reflecting on the proportionalReflecting on the proportionalreasoning unitreasoning unit

What effective teaching strategies used by Raj What effective teaching strategies used by Raj could you consider adopting in your department?could you consider adopting in your department?

How has the structure of this unit and the Year 8 How has the structure of this unit and the Year 8 unit suggested a model for tracking through a unit suggested a model for tracking through a mathematical topic across two or more years?mathematical topic across two or more years?

How has this unit suggested ways of making How has this unit suggested ways of making further links between strands of the yearly further links between strands of the yearly teaching programmes?teaching programmes?

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

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