1. generalising a mean question
how can a strip of cardboard be cut into three pieces so that one piece is the average length of the other two?

2. there are two strips of cardboard
is it always possible to cut one of them so that one of the pieces is the average length of the other two?
David Wells

3. generalising a division by 4 property

4. dividing by 5 1 4 2 8 5 7 5 2 1 4 2 3

5. dividing by 4 2 5 1 2 8 4 2 1 3

6. dividing by 4 1 5 3 8 4 6 4 2 1 3 1 2

7. you might notice some relationship between what you get and the original number (dividing by 4)
find other six digit numbers that ‘work’ in this way, when dividing by 4,
numbers that start with: 1 2 3 4 5 7 9
[done 6 and 8 already]

8. 8 2 0 2 5 3 1 6 4 5 5 6 9 6 dividing by 8 there is another one…
starting with a 3

9. generalising star polygons

10. 9/4

11. 12/5

12. generalising: reflect in some simple lines
what happens when you reflect a point, any point, in:
x = 3
y = x
y = – x
x + y = 4
y = 2x
what happens when you rotate a point, any point, through 180o about
the origin
another point?
90o
generalising: reflect in some simple lines

14. (0, 2) → (2, 4)
(0, 0) → (4, 4)
(2, 1) → (3, 2)
(– 2, 2) → (2, 6)

15. (4, 3) → (0, 5)
(5, 0) → (– 3 , 4)
(1, 2) → (1, 2)
(– 7, 1) → (5, – 5 )

16. generalising: add and take– +

17. what might you write (with numbers) for these?

18. for these?

19. 1.)
2.)
– 4 =
– 8 =
– 15 =
1 + 2 – 3 =
7 + 8 – 9 =
– 14 =
3.)
4.)
1 – 2 – =
6 – 7 – =
15 – 16 – = = = =
work out the results for these calculations

20. 5.)
6.)
1 – =
7 – =
12 – =
1 + 2 – =
6 + 7 – =
– =
7.)
8.)
2 + 4 – =
7 + 9 – =
– = = = =
work out the results for these calculations

21. 9.)
10.)
1 – =
4 – =
7 – =
2 – = = = = =
11.)
12.)
– 7 =
– 16 =
– 13 =
– 6 =
your own…

22. generalising: add and take
what is the pattern? give a few more examples (at least three) and try to find a general rule (using n):
1.)
2.)
3.)
1 + 2 – 3 =
7 + 8 – 9 =
– 12 =
– 4 =
– 10 =
– 15 =
1 – 2 – =
6 – 7 – =
15 – 16 – =
4.)
5.)
6.) = = =
1 + 2 – =
7 + 8 – =
– =
1 – =
6 – =
9 – =
7.)
8.)
9.)
2 + 4 – =
7 + 9 – =
– = = = =
1 – =
4 – =
7 – =

23. which question is this?
what do the diagrams show?