1. Good tasks, good questions, good teaching, good learning …. Anne Watson Leeds PGCE Feb 2007

2. Decimals! 10% of 23 2.3 20% of 23.23

3. Teaching context All learners generalise all the time It is the teacher’s role to organise experience It is the learners’ role to make sense of experience

4. (a)P = (1, -1) (b)P = (-2, -4) (c) P = (-1, -3) (d) P = (0, -2) (e) P = (½, -1½ ) (f) P = (-1½, -3½) (g) P = (0, 0) (h) P= (-2, 2) Taxicab distances Let A =(-2, -1)

5. Phrases we are not going to use Today we are going to do page 93 … Then they did the exercise … I gave them a worksheet … They practised …

6. Gradient exercise 1: (4, 3) & (8, 12) (-2, -1) & (-10, 1) (7, 4) & (-4, 8) (8, -7) & (11, -1) (6, -4) & (6, 7) (-5, 2) & (10, 6) (-5, 2) & (-3, -9) (-6, -9) & (-6, -8) (8, 9) & (2, -9) (7, -8) & (-7, 5) (-9, -7) & (1, 4) (-4, -3) & (4, -2) (2, -5) & (-3, -7) (1, 6) & (-1, -3) (-1, 0) & (5, -1) (-3, 5) & (-3, 2)

7. Gradient exercise 2: (i) (4, 3) & (8, 12)(ii) (-2, -3) & (4, 6) (iii) (5, 6) & (10, 2)(iv) (-3, 4) & (8, -6) (v) (-5, 3) & (2, 3)(vi) (2, 1) & (2, 9) (vii) (p, q) & (r, s)(viii) (0, a) & (a, 0) (ix) (0, 0) & (a, b)

8. Gradient exercise 3: (4, 3) & (8, 12)(4, 3) & (4, 12) (4, 3) & (7, 12)(4, 3) & (3, 12) (4, 3) & (6, 12)(4, 3) & (2, 12) (4, 3) & (5, 12)(4, 3) & (1, 12)

9. What do you see?

10. Use of controlled variation 4 pens plus 5 pencils cost £2.60 4 pens plus 2 pencils cost £2.00 5 oranges plus 3 apples cost £2.36 5 oranges plus 1 apple cost £2.12 8 stamps plus 5 envelopes cost £3.90 8 stamps plus 4 envelopes cost £3.60

11. Controlling variation and using layout to show structure sin 2 x + cos 2 x = 1 2 sin 2 x + 2 cos 2 x= 2 3 sin 2 x + 3 cos 2 x = 3 4 sin 2 x + 4 cos 2 x = 4 e x sin 2 x + e x cos 2 x = e x cosx sin 2 x + cos 3 x = cosx

12. Giving choice; learners’ examples Multiply each of the terms in the top row by each of the terms in the bottom row in pairs: x – 1x + 1x + 2 x + 3 Add some more options of your own

13. Answers worth comparing Simplify these: 6/10 18/20 6/8 14/16 Now simplify these: 15/25 45/50 15/2035/40 Compare the answers

14. Sorting 2x + 13x – 32x – 5 x + 1-x – 5x – 3 3x + 33x – 1-2x + 1 -x + 2x + 2x - 2

15. Sorting processes Sort into two groups – not necessarily equal in size Describe the two groups Now sort the biggest pile into two groups Describe these two groups Make a new example for the smallest groups Choose one to get rid of which would make the sorting task different

16. Sorting grids +ve y-intercept -ve y-intercept Goes through origin +ve gradient -ve gradient

17. Sorting trees What is the x-coefficient? 123

18. Comparing In what ways are these pairs the same, and in what ways are they different? 4x + 8 and 4(x + 2) Rectangles and parallelograms Which is bigger? 5/6 or 7/9 A 4 centimetre square or 4 square centimetres

19. Ordering Put these in increasing order: 6√2 4√3 2√8 2√9 9 4√4

20. Put these in order of …… x √2 e x/2 3 √ x 2 2 x x -2/3 x √2 x 3/2 3 √ x 2 x 2sin x x -2/3

21. Arguing about … Anne says that when a percentage goes down, the actual number goes down - Is this always, sometimes or never true? John says that when you square a number, the result is always bigger than the number you started with - Is this always, sometimes or never true?

22. Characterising Which multiples of 3 are also square numbers? Which quadratic curves go through (0,0)?

23. Needing harder methods Find a number half-way between: 28 and 34 2.8 and 3.4 38 and 44 -34 and -28 9028 and 9034.0058 and.0064

24. Needing harder methods Find a number half-way between: and

25. Using numbers as placeholders 1 x 7 1 x 7 1 x 7 … 7 2 7 3 7 4 3 x 7 3 x 14 3 x 21… 7 8 7 15 7 22 9 x 14 … 21

26. Varying order … 2x – 3 x + 4(5x + 2)/2

27. Varying order …. adding 1 dividing by 1 subtracting 1 multiplying by 1 substitute n for 1 and find values for n which change the order

28. … and another Find a quadratic whose roots have a difference of three … find another

29. Purposeful textbook tasks

30. Summary of key ideas Exercise design: expectation, surprise, practice Control variation: a lot, a little, what? Interplay of examples and generalisation Visual impact Complexifying Choice Making up examples Comparing answers Sorting Ordering Arguing about … Characterising Leading into harder methods Numbers as placeholders … and another