1. Teaching Secondary Maths in Worcestershire Day 2
18th January 2007

2. 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.

3. Programme:
Developing skills in geometric reasoning
Coffee
Promoting thinking
Lunch
Data handling - leading to coursework
Tea/Coffee
Preparing pupils for exams

4. Geometrical Reasoning
Session 1

5. 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.

6. 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.

7. 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.

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

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

10. 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

11. 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!

12. 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.

13. 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’

14. 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.

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

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

17. 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

18. 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)

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

20. 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

21. 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

22. Mathematical proof
START
What am I assuming?
reasoning
NEW RESULT

23. Using ICT
Cabri
Geometer’s sketch pad

24. Using ICT Geogebra tools drawing pad algebra window input commands
symbols

25. 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