Key Stage 3 National Strategy Planning and teaching mathematics 2 Geometry, ratio and proportion, and problem solving

Key Stage 3 National Strategy Session 9 Reflecting on lessons taught and improving questioning strategies

Key Stage 3 National Strategy Objectives for session 9  To reflect on and share successful aspects of lessons planned in previous training  To consider how questioning can be used to engender effective discussion 9.0

Key Stage 3 National Strategy Planning and teaching mathematics 2 Day 3Planning and teaching geometrical reasoning Session 9Reflecting on lessons taught and improving questioning strategies Session 10Geometrical reasoning 1 Session 11Geometrical reasoning 2 Session 12Planning to teach geometry Day 4Planning and teaching ratio and proportion and problem solving Session 13Ratio and proportion 1 Session 14Ratio and proportion 2 Session 15Teaching problem solving 1 Session 16Teaching problem solving 2 Session 17Next steps 9.1

Key Stage 3 National Strategy Questioning and discussing  as many pupils as possible take part  questions are used to assess pupils’ progress against the teaching objectives and are framed to allow understanding to be assessed  pupils are given time to think and discuss their ideas before answering 9.2a How can lessons be planned so that:

Key Stage 3 National Strategy  open and closed questions are planned and adjusted to encourage pupils to: – recall facts – apply facts – hypothesise and predict – design and compare procedures – interpret results – apply reasoning 9.2b Questioning and discussing How can lessons be planned so that:

Key Stage 3 National Strategy Objectives for session 9  To reflect on and share successful aspects of lessons planned in previous training  To consider how questioning can be used to engender effective discussion 9.0

Key Stage 3 National Strategy Geometrical reasoning 1 Session 10

Key Stage 3 National Strategy Objectives for session 10  To look at approaches to developing pupils’ visualisation and geometrical reasoning skills  To consider progression towards geometric proof  To increase familiarity with the Framework for teaching mathematics: Years 7, 8 and

Key Stage 3 National Strategy Shape and space: Year 9 objectives  Distinguish between conventions, definitions and derived properties  Distinguish between demonstration and proof (Year 9 extension) 10.1

Key Stage 3 National Strategy  Labelling  Notation 10.2 ABABABAB A AABCBCAABCBC B ca ca A b C Examples of conventions t

Key Stage 3 National Strategy Examples of definitions  Corresponding angles lie on the same side of a transversal and on corresponding sides of parallel lines. 10.3a

Key Stage 3 National Strategy  Angle ACD is an exterior angle of ABC. ABCD 10.3b Examples of definitions

Key Stage 3 National Strategy  Perpendicular lines intersect at right angles. 10.3c Examples of definitions

Key Stage 3 National Strategy Towards proof in Key Stage 3 Stage 1Convince yourself (mental justification) Stage 2Convince a friend (oral justification) Stage 3Convince a pen-friend (informal written justification) Stage 4Convince your mathematics teacher (more formal written justification) (Adapted from ‘Can you prove it?’ by Sue Waring, The Mathematical Association) 10.4

Key Stage 3 National Strategy Derived properties 10.5a Stage 2 Vertically opposite angles are equal Stage 1

Key Stage 3 National Strategy Stage 4 + = 180 o Stage 3 y z w x + = 180 o So = x + y = 180 o (angles on a straight line) y + z = 180 o (angles on a straight line) So x = z 10.5b Derived properties

Key Stage 3 National Strategy Objectives for session 10  To look at approaches to developing pupils’ visualisation and geometrical reasoning skills  To consider progression towards geometric proof  To increase familiarity with the Framework for teaching mathematics: Years 7, 8 and

Key Stage 3 National Strategy Geometrical reasoning 2 Session 11

Key Stage 3 National Strategy Objectives for session 11  To explore the connections between loci and constructions  To consider a variety of activities and resources to develop pupils’ visual and geometrical reasoning skills 11.0

Key Stage 3 National Strategy People geometry One person is the mover. Two are checkers. Two are instructors. Rules The instructors are the only ones allowed to speak or have ideas! The checkers are the only ones allowed to measure. They may use two balls of string. The mover and checkers must wait for instructions. 11.1

Key Stage 3 National Strategy Objectives for session 11  To explore the connections between loci and constructions  To consider a variety of activities and resources to develop pupils’ visual and geometrical reasoning skills 11.0

Key Stage 3 National Strategy Planning to teach geometry Session 12

Key Stage 3 National Strategy Objectives for session 12  To write and review a geometry lesson plan  To begin to consider how a sequence of related lessons is planned 12.0

Key Stage 3 National Strategy Ratio and proportion 1 Session 13

Key Stage 3 National Strategy Objectives for session 13  To explore ratio and proportion as key mathematical ideas in both number and algebra, with applications to other subjects  To analyse a lesson on ratio and proportion  To examine links within the yearly teaching programmes and related examples in the supplement 13.0

Key Stage 3 National Strategy Recording ratios of length 13.1 R lYlY—lRlRlYlY—lRlR lRlR—lYlYlRlR—lYlY 5 55—2255— —5522— —2255— —2211— —5522—555 Y ratio of l Y to l R = 5 : 2 l Y =  l R (or 2  l R ) ratio of l R to l Y = 2 : 5 l R =  l Y ==

Key Stage 3 National Strategy Some key points about ratio  Ratio involves division of one number by another –Sometimes the result will be a whole number –Sometimes the result has to be left as a fraction or mixed number (or converted to an equivalent decimal or percentage) 13.2a

Key Stage 3 National Strategy Some key points about ratio  There are several ways of expressing a ratio –colon notation –the fraction or division line (reading ‘/’ as ‘÷’) –multiplication (reading ‘  ’ as ‘of’)  The inverse of the ratio a : b is b : a The associated fractions are and its reciprocal a—b b—a 13.2b

Key Stage 3 National Strategy Objectives for session 13  To explore ratio and proportion as key mathematical ideas in both number and algebra, with applications to other subjects  To analyse a lesson on ratio and proportion  To examine links within the yearly teaching programmes and related examples in the supplement 13.0

Key Stage 3 National Strategy Ratio and proportion 2 Session 14

Key Stage 3 National Strategy Objectives for session 14  To consider methods for solving problems involving ratio and proportion  To consider how mathematical problems and methods can be simplified or made more challenging to meet the needs of different pupils  To make links between ratio and proportion and enlargement and similarity 14.0

Key Stage 3 National Strategy Different ways of solving a proportion problem  Informal scaling methods  Finding the scale factor  Unitary method  Algebraic method 14.1

Key Stage 3 National Strategy Solving ratio and proportion problems  It is helpful to think carefully about how to start a problem and to compare different methods  Informal scaling enables pupils to work within their understanding, provided the numbers are amenable  Scaling, unitary and algebraic methods are more direct and work for any numbers 14.2

Key Stage 3 National Strategy Circle and diameter A A

Key Stage 3 National Strategy 14.4 A Circle and diameter 2

Key Stage 3 National Strategy Circle and diameter A

Key Stage 3 National Strategy Circle and diameter A

Key Stage 3 National Strategy Circle and diameter A

Key Stage 3 National Strategy Circle and diameter A x x x x

Key Stage 3 National Strategy Circle and diameter 2 x x x x 14.4 A

Key Stage 3 National Strategy Objectives for session 14  To consider methods for solving problems involving ratio and proportion  To consider how mathematical problems and methods can be simplified or made more challenging to meet the needs of different pupils  To make links between ratio and proportion and enlargement and similarity 14.0

Key Stage 3 National Strategy Teaching problem solving 1 Session 15

Key Stage 3 National Strategy Objectives for session 15  To consider the nature of problem solving in Key Stage 3 and the implications for planning and teaching  To become familiar with the examples for using and applying mathematics in the Framework supplement 15.0

Key Stage 3 National Strategy Problem solving  Using and applying mathematics already taught in a problem-solving context  Problems or starting points that enable pupils to explore and investigate new mathematics 15.1

Key Stage 3 National Strategy Objectives for session 15  To consider the nature of problem solving in Key Stage 3 and the implications for planning and teaching  To become familiar with the examples for using and applying mathematics in the Framework supplement 15.0

Key Stage 3 National Strategy Teaching problem solving 2 Session 16

Key Stage 3 National Strategy Objective for session 16  To explore questioning strategies: – to engage pupils in problem solving – to probe pupils’ understanding 16.0

Key Stage 3 National Strategy Probing questions: Year 6  What percentages can you easily work out in your head? Talk me through a couple of examples.  When calculating percentages of quantities, what percentage do you usually start from? How do you use this percentage to work out others?  Are there any percentages that you cannot work out? 16.1a Understand percentage as the number of parts in every 100, and find simple percentages of small whole- number quantities.

Key Stage 3 National Strategy  50% of the numbers on this 1–100 grid are even. How would you check?  Give me a question with an answer of 20%.  ‘To calculate 10% of a quantity you divide it by 10. So to find 20% you must divide by 20.’ What is wrong with this statement? Probing questions: Year b Understand percentage as the number of parts in every 100, and find simple percentages of small whole- number quantities.

Key Stage 3 National Strategy Probing questions: starting points  How do you know …? What do you look for?  How do you go about …?  What clues do you use when …?  Make up some questions that …  Give me a … that is the same as …  How did you do it?  Present a statement and ask: – Is it always true? – Why is it false? 16.2

Key Stage 3 National Strategy Objective for session 16  To explore questioning strategies: – to engage pupils in problem solving – to probe pupils’ understanding 16.0

Key Stage 3 National Strategy Next steps Session 17

Key Stage 3 National Strategy Objectives for session 17  To reflect on what has been covered on the course  To consider aspects of planning and teaching for future development 17.0

Key Stage 3 National Strategy Planning and teaching mathematics  Algebraic reasoning  Approaches to calculation  Geometrical reasoning  Proportional reasoning  Problem solving 17.1

Key Stage 3 National Strategy Objectives for session 17  To reflect on what has been covered on the course  To consider aspects of planning and teaching for future development 17.0