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

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

2. 2 × 4 = 8.
Double 2 × 4 = 8
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 ?
5 × 4 and 3 × 4
6 × 4 and 2 × 4
7 × 4 and 1 × 4
5 fours
are 20
3 fours
are 12
6 fours
are 24
2 fours
are 8
Facts within facts

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

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
are 20
3 fours
are 12
6 fours
are 24
2 fours
are 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, students could double one factor
(3 in this case) to get 6 × 4. They could double the other
factor to get 3 × 8.
For some facts, they could halve one and double the other.
Facts within facts

19. Going further: ideas and tips
Teaching tips
Drawing the arrays on the grid provides a visual image of how
the dimensions change 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

20. Going further: ideas and tips
Build up a class list of multiplication
facts on the multiplication grid generated
by doubling and partitioning.
Facts within facts

21. Going further: ideas and tips
Students may also explore the use of
halving and doubling to work out tricky
facts such as 7 lots of 8 and related
division facts.
For example, if I know 7 fours is 28, then I
can double the fours to get 7 eights.
Double the product of 28 gives 56, which
is the product of 7 and 8.
Facts within facts