1. Multiplication and Division

2. The meaning of multiplication??
3 x 4 = 12
3 x 4 = 12
Mathematically correct
3 ‘lots of’ 4
make 12
3 multiplied 4 times makes 12
What language would you use to communicate multiplication? – Discuss with a partner - lots of, times, groups
What visual models could you use? – Discuss - beads, number lines, arrays

3. Count in multiples supported by concrete objects in equal groups
Repeated addition

4. Multiplication as repeated addition
+ 4
+ 4
+ 4
3 x 4 = 12
+ 4
+ 4
+ 4
- Can you think of an alternative method? - multiplication is commutative, can be done in any order. 20 4 8 12
3 x 4 = 12

5. Bar model

6. The commutative law of multiplication
Arrays
Link with inverse, division - triangles
The commutative law of multiplication

7. The commutative law of multiplication
+ 4
+ 4
+ 4 12 20 4 8 12
3 x 4 = 12
+ 3
+ 3
+ 3
+ 3 12 20 3 6 9 12
4 x 3 = 12

8. Scaling
This can be generalised to include any multiplier including those less than one – i.e. making smaller
X 1/3
3 times
as tall 8

10. Grid Method
13 x 4
Progression in manipulatives
X 4 so 4 rows needed

11. Grid Method
35 x 7

12. Column Multiplication
Expanded method
Short method

13. More than single digits? 18 x 13 =10 18 18 8 10
100 80 13 13 3 30 24

14. Progressing towards the standard algorithm
1 0 8
1 0
1 0 0
8 0 3
3 0
2 4

15. Long Multiplication

16. Expectations – Year 3 & 4
Year 3 – 3, 4 and 8 multiplication facts and moving towards formal multiplication of up to 2 digit x 1 digit
Year 4 – 12 x 12 multiplication facts, multiplying together three numbers and multiply 2 digit and 3 digit numbers by a 1 digit number using formal written layout
Order of times tables
10, 5, 2, 4, 3, 6, 9, 8, 7, 11, 12
2 minute tables – daily
Y6 On screen test in summer 2017

17. Expectations – Year 5 & 6
Year 5 & 6 – multiply numbers up to 4 digits (including decimals) by a 1 or 2 digit number using a formal written method, including long multiplication for 2 digit numbers
2 minute tables – daily
Y6 On screen test in summer 2017

18. Division 12 ÷ 4 How do you picture this?
Sharing: number of groups is known
quantity in each group is unknown
Grouping: number of groups is unknown
quantity in each group is known
How do you picture division? What language would you use? – discuss

19. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

20. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

21. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

22. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

23. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

24. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

25. Looking at progression in division
Practical Problems – division as sharing
6  2 =
Six teddy bears are shared between two children.
How many teddy bears does each child get?

26. Looking at progression in division
Practical Problems – division as sharing
6  2 = 3
Six teddy bears are shared between two children.
How many teddy bears does each child get?

27. Looking at progression in division
Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

28. Looking at progression in division
Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

29. Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

30. Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

31. Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

32. Practical Problems – division as grouping
12  4 =
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

33. Practical Problems – division as grouping
12  4 = 3
Twelve teddy bears go for a picnic. Four bears can sit at each picnic table. How many picnic tables do they need?

34. 8  2 = 4 Example of Division Two frogs can sit on a lily pad.
If I have 8 frogs how many lily pads will I need so that they can all sit on lily pads?
Sharing or grouping?
8  2 = 4
Discuss – is this problem sharing or grouping?

35. 20  5 = 4 Example of Division There are 20 bones to give to 5 dogs.
How many bones should each dog have?
Sharing or grouping?
20  5 = 4
Discuss – is this problem sharing or grouping?
Have a go …… can you think of a grouping problem - share

36. An image for 56  7
The array is an image for division too
5 6 7 8
5 6 7 8
The key point here is that, by arranging the groups you remove into an array, grouping and sharing can be seen as ‘two sides of the same coin’ and not mutually exclusive.
It relates to 8 x7

37. 363 ÷ 3 = 1 2 1
3 6 3 3
Division with no exchange or “re group” and no remainder
Highlight that you are going to talk through division as grouping
Emphasise that when we make a group of 3 in the hundreds column it is one group of 3 but the counters are hundreds and we can only call it one group of 3, within the context of it being in the hundreds column. The digit is multiplied by the column heading, so the one has a value of 100 make reference back to the place value in session 1
Now model as Sharing
Talking through, in the first column, the counters shared between 3 means that they have 1 counter each
Talk this through for each of the columns
Identify that children need to see the model as grouping in order to more fluently use their tables.

38. 364 ÷ 3 =
3 6 4 3

39. 364 ÷ 3 = 1 2
rem 1
3 6 4 3
If there are some left over, children are quite happy with this and you could introduce the language of ‘remainder’ together with some way of recording this.
N.B. There is the possibility, at a later stage, to exchanged the one for tens 0.1s. Might want to mention this here(?)

40. 345 ÷ 3 = 1 1 5
3 4 5 3 1

41. 3 2 0 6 1 3 8 138 Short Division 1 1 23 6 In the array
2 0 1 6
1 3 8 1 23
In the array
We can explicitly see 23 six times, 6 rows of 23
This is the sharing model, a 1/6th of 138 is 23
We can divide the array up into six parts and there are 23 in each part
But we cannot explicitly see
Twenty three times in the place value counters
This is the grouping model – How many groups of six are there in twenty three – the image doesn’t lend itself to this. We cannot see 23 columns of 6 6
138 41

42. Long Division

43. Expectations Year 3 & 4
Year 3 – recall and use division facts for the 3, 4 and 8 multiplication tables
Year 4 – recall division facts for multiplication tables up to 12 × 12 and divide mentally

44. Expectations Year 5 & 6
Year 5 - divide numbers up to 4 digits by a1 digit number using the formal written method of short division and interpret remainders
Year 6 – divide numbers up to 4 digits by a 2 digit whole number using the formal written method of long division,