Teaching Secondary Maths in Worcestershire Day 2 18th January 2007

Objectives of the day: To have an increased awareness of the progression in skills across key stages 3 and 4. To be more confident in the classroom, with regard to planning structured lessons and teaching KS3 and GCSE classes. To be more effective in teaching, marking and moderating GCSE coursework tasks.

Programme: 9.00-10.20 10.20-10.40 10.40-12.00 12.00-1.00 1.00-2.20 2.20-2.40 2.40-4.00 Developing skills in geometric reasoning Coffee Promoting thinking Lunch Data handling - leading to coursework Tea/Coffee Preparing pupils for exams

Geometrical Reasoning Session 1

Developing skills in geometrical reasoning Identify the progression in construction skills through KS1, 2 and 3. Useful starters and plenary activities. Work towards a formal proof. Link ICT to geometrical reasoning.

Progression in constructions skills Each card contains a skill taken from the Primary/Secondary Framework Place the cards in the order of progression. Link them to the appropriate year.

Reflection Where does your teaching start in the progression? Is what you are teaching appropriate to the target levels of the pupils? What do you need to do to check? Make a note on your action points sheet.

Oral and mental starters Activities to develop visualisation skills and mental geometric reasoning visualisations true or false draw that shape

Visualisations and duals dual of a cube Year 9 booster pack Dialogue and Reasoning dvd

True or false? T F - area F - units F - reversed F - perimeter 1 The rectangle with sides 5cm and 6cm, has an area of 30sqcm and perimeter of 22cm. 2 The rectangle with sides 3cm and 2cm, has an area of 32sqcm and perimeter of 10cm. 3 The rectangle with sides 6cm and 4cm, has an area of 24sqcm and perimeter of 20cm. 4 The rectangle with sides 2cm and 10cm, has an area of 20cm and perimeter of 24sq cm. 5 The rectangle with sides 5cm and 4cm, has an area of 18sqcm and perimeter of 20cm. 6 The rectangle with sides 2cm and 3cm, has an area of 6sqcm and perimeter of 5cm. T F - area F - units F - reversed F - perimeter

Draw that shape 7 You will get 4 sentences. They will be revealed one by one. Predict the number of sentences you think you will need to be able to draw the correct shape, and show your number. If you say 2, you must draw and show a shape after 2 sentences. Whiteboards and pens ready!

Draw that shape 7 Shape 1  I am an arrowhead. I have 1 line of symmetry. 2 of my sides are 1 cm long. I have 1 internal dot.

Oral and mental starters start with basic principles build visualisation skills build vocabulary build sentence construction build confidence support with practical frameworks lead on to deductive reasoning by looking for ‘because’

Working towards a geometrical proof Section 3- Yearly teaching programmes Year 7- page 7- geometrical reasoning- lines, angles and shapes Year 8- page 9 Year 9- pages 11 and 13 Jot down any words that demonstrate geometrical reasoning skills.

Examples of conventions c b B a C A B C Labelling Notation t A B 10.2

Examples of definitions Corresponding angles on parallel lines lie on the same side of a transversal and on corresponding sides of parallel lines.

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

Prove that vertically opposite angles are equal Convince yourself (mentally) Convince a friend (verbally) Convince a pen-friend (jottings) Convince your maths teacher (formal proof)

Derived properties Stage 1 Stage 2 Vertically opposite angles are equal Stage 1 Stage 2 10.5a

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

Try this at home Prove that the sum of the interior angles of an n-sided polygon add up to (n-2) x 180° Use the four stages:- Convince yourself Convince a friend Convince a pen-friend Convince your maths teacher

Mathematical proof START What am I assuming? reasoning NEW RESULT

Using ICT Cabri Geometer’s sketch pad

Using ICT Geogebra tools drawing pad algebra window input commands symbols

Action points Know the progression of geometrical reasoning in the Framework Be able to introduce oral and mental starters to support development of reasoning Use formal conventions and definitions to support convincing proofs Use ICT to support development of proof