Top Drawer Teachers The Australian Association of Mathematics Teachers (AAMT) Inc. http://topdrawer.aamt.edu.au

1. If I know 2 × 4 = 8
what else do I know?
Facts within facts

2. 2 × 4 = 8 double it gives 4 × 4 = 16
Double 4 × 4 = 16 gives 8 × 4 = 32
If I know 8 × 4 = 32
I know 4 × 8 =32
2 fours
are 8
4 fours
are 16 8 fours
are 32
Facts within facts
If I know 2 × 4 = 8
what else do I know?

3. What other facts can you see in 8 × 4 ?
Facts within facts

4. What can I see in 8 × 4 ?
Within 8 × 4 = 32 I can see:
5 × 4 and 3 × 4
6 × 4 and 2 × 4
7 × 4 and 1 × 4
5 fours
is 20
3
fours
is 12
6 fours
is 24
2
fours
is 8
Facts within facts

5. What can I see in 8 × 4 ?
Within 8 × 4 = 32 I can see:
5 × 4 and 3 × 4
6 × 4 and 2 × 4
7 × 4 and 1 × 4
This can help work out:
10 × 4 (double 5 × 4)
5 fours
is 20
3
fours
is 12
6 fours
is 24
2
fours
is 8
Facts within facts

6. What can I see in 8 × 4 ?
If I double 8 × 4 = 32
I get 8 × 8 = 64
What can this help me
to work out?
5 fours
is 20
3
fours
is 12
6 fours
is 24
2
fours
is 8
Facts within facts

7. What else can I see?
Within 8 × 4 = 32 I can see:
5 × 4 and 3 × 4
Facts within facts

8. What else can I see?
Within 8 × 4 = 32 I can see:
5 × 4 and 3 × 4
4 × 5 = 20 20 ÷ 5 = 4
4 × 3 = 12 12 ÷ 3 = 4
Facts within facts

9. What else can I see?
Within 8 × 4 = 32 I can see:
6 × 4 and 2 × 4
Facts within facts

10. What else can I see?
Within 8 × 4 = 32 I can see:
6 × 4 and 2 × 4
Facts within facts
4 × 6 = 24 24 ÷ 6 = 4
4 × 2 = 8 8 ÷ 2 = 4

11. What else can I see?
Within 8 × 4 = 32 I can see:
7 × 4 and 1 × 4
Facts within facts

12. What else can I see?
Within 8 × 4 = 32 I can see:
7 × 4 and 1 × 4
Facts within facts
4 × 7 = 28 28 ÷ 7 = 4
4 × 1 = 4 4 ÷ 1 = 4

13. What else can I see?
If I double 8 and halve 4 what do I get?
Facts within facts

14. What else can I see?
If I double 8 and halve 4 what do I get?
Facts within facts
16 × 2 = 32

15. What else can I see?
If I double 8 and halve 4 what do I get?
Facts within facts
16 × 2 = 32
Knowing this I also know:
2 × 16 = 32
32 ÷ 16 = 2
32 ÷ 2 = 16

16. × 1 2 3 4 5 6 7 8 9 10
1 4
2 8
3 12
4 4 8 12 16 20 24 28 32 40
5 20
6 24
7 28
8 32 64
9
10 40
Facts within facts

17. × 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
Facts within facts

18. Going further ideas and tips
• In pairs, students choose a known fact and use
doubles and partitioning to record some related
facts.
For example, given 3 × 4, doubling one factor
(3 in this case) to get 6 × 4, doubling and
halving both factors to get 3 × 8.
Teaching tips
Drawing the arrays on the grid provides a visual
image of how the dimensions when one factor
is doubled.
It can also show ways to partition the array.
Recording the product on the multiplication grid
provides the link between the visual and the
symbolic representation.
Facts within facts

19. Going further ideas and tips
• Build up a class list of multiplication facts
on the multiplication grid generated by
doubling and partitioning.
• Students may also explore use of halving and
doubling to work out tricky facts such as 7
eights and related division facts.
For example, if I know 7 fours is 28 then I
double the fours get 7 eights. Double the
product of 28 to get 56 (i.e. 7× 8).
Facts within facts