1. Aims of the Session
To build understanding of mathematics and it’s development throughout KS2
To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods)
To enhance subject knowledge of the pedagogical approaches to teaching mathematics

2. Division
At the heart of success of this topic is clearly mastery of times tables. The more fluent a pupil is at their tables, the easier they will find division.
Practice those times tables at every opportunity. Remember:
Lining up to assembly (and hopefully other teachers will follow suit).
Getting changed for PE, etc.
Parents! They can really support the regularity of practice (daily?)

3. Grouping and Sharing 12 ÷ 3 = 4
Grouping – we know how many are in each group but not how many groups there will be. The answer is the number of groups.
Sharing - we know how many groups there are but not how many are in each group. The answer is the number in each group.
Sharing equally is not necessarily understood by children when they use the word sharing! Grouping and sharing are introduced physically in Y1 in the new curriculum – both models
Grouping

4. Bar Model?
I had 32 sweets and I shared them between 4 children. How many did each child get?
I had 32 chocolate bars and they came in packs of 4. How many packs did I get?

5. Activities to support multiplication and division on a number line
Grab
Progression ideas:
Size of objects
Choice of numbers
Scoring
Remainders
Representation
Play grab against each other.
Grab a handful of stuff – count it. Then mark against divisors eg 2,5,10
Who wins?
52 divided by 4 – Do the children partition into 40 and 12?

6. Ella has 48 plasticine legs to make animals for a display.
How many cows could she make?
How many beetles could she make?
Where is the division language in these questions? Apply RUCSAC to this problem – does it help?
Children have to understand the grouping and sharing models in order to identify the maths involved.
Can children even write a number sentence for this question – or just do it without appreciating the division
(mustn’t have teaching giving it away as 48/4)
How many spiders could she make?

7. Building the Journey
Year 3 Pupils can derive associated division facts e.g. if 6 ÷ 3 = 2, then 60 ÷ 3 = 20
Pupils develop reliable written methods for division, progressing to the formal written methods of short division.
Year 4 Pupils can derive associated division facts e.g. if 28 ÷ 7 = 4, then 2800 ÷ 7 = 400
Pupils practise to become fluent in the formal written method of short division
with exact answers (appendix 1 - how do you explain this!).
Year 5 Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context (appendix 1 - how do you explain this!).
Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, e.g. in scale drawings or in converting between metric units.
Year 6 Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context.
Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context.
Solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts.
Solve problems involving similar shapes where the scale factor is known or can be found.

8. An image for 56  7 Either: How many 7s can I see? (grouping) Or:
If I put these into 7 groups how many in each group? (sharing)
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

9. 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

10. Reflection
In my eyes, there is one major part missing (regarding division) in the build up of division - a step which shouldn't be missed!
Autumn?
Concentration of formal addition and subtraction.
Division via arrays and mental partitioning.
Spring?
Concentration of formal multiplication
Division – as per Autumn – harder numbers?
Recognise the number operation.
Summer?
Introduce the formal algorithm with Dienes / Place Value counters.
(Keep division in consistent with the year group's times tables) T
Autumn…. 52/4 by partitioning
Summer – mover/scribe model… this gets developed in year 4 – so priority given to modelling short division, not necessarily recording.

11. Reflection
In my eyes, there is one major part missing (regarding division) in the build up of division - a step which shouldn't be missed!
Year 3
Autumn? Spring? Summer?
60 ÷ 3
80 ÷ 4
63 ÷ 3
48 ÷ 3
92 ÷ 4
(Keep division in consistent with the year group's times tables)
Discuss – how could you tackle these in each term?
Autumn – lots of drawing of arrays in problems
Spring – chunking (include remainders)
Summer – tentative illustration of “short division” – or stick with chunking – schools need to make the right decision for their students.

12. 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
“I am looking in the 100s column, how many groups of 3 100s can I make”

13. 364 ÷ 3 =
3 6 4 3

14. 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(?)

15. 345 ÷ 3 = 1 1 5
3 4 5 3 1

16. Year 4
The journey is now about fluency with short division. Although not explicit, three-digit divided by one-digit seems a sensible goal by the end of the year. There should be exact answers (no remainders). What if pupils "master" the process in the autumn term?
Do you have "tricks" as a teacher to ensure there are no remainders?!
462 ÷ 2
725 ÷ 5
537 ÷ 3
474 ÷ 6
738 ÷ 9 H T U
100 10 1
On last two – more than acceptable to put a 0 in H column

17. Year 5
Division moves up to four digits, with the emphasis on manipulating the remainder. Manipulation includes:
simply recognising it exists!
understanding how to get a decimal answer (missing from appendix 1)
understanding how to get a fractional answer.
Note: the decimal and fractional answer could easily be taught "as a process", but the emphasis must be on understanding.
Again the pupil who "masters" the manipulation quickly, must have their division enriched by appropriate problem solving opportunities.

18. Year 5 Dealing with remainders (needs differentiating carefully)
These three examples have been chosen to illustrate various outcomes that might happen. Scaffold the manipulation of the remainder carefully over the year, as and when different groups are ready for the next step.
e.g.
7143 ÷ 5
3466 ÷ 8
3416 ÷ 6 U T H Th
1000
100 10
Tents!
Sleeping out on the grass!
School coaches for a trip. 1
Dividing by 6 happens when putting eggs into boxes?!

19. Year 6
Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context.
Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context.
(Note: as a mathematician, I never use long division, and do not see its value in the new National Curriculum... but you have to teach it!)
Careful then writing recurring decimals...
Mike Askew says you cannot teach long division with conceptual understanding but they should have enough understanding of division by now to take the process.

20. Year 6
Solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts.
This is ratio!
I always think the most natural way to introduce this is through "mixing paint"... let you pupils take ownership of their learning...

21. (Hidden) Applications of Ratio:
If 3 pencils cost 45p, how much did one pencil cost?
If 2 pencils cost 60p, how much would 5 pencils cost?
If 5 pencils cost 70p, how many pencils could I buy for £2.10?
Ingredients to make 16 gingerbread men
180 g flour
40 g ginger
110 g butter
30 g sugar
How much of each ingredient would you need to make gingerbread men?
Link at all stages with the bar model

22. Year 6
Solve problems involving similar shapes where the scale factor is known or can be found.
Language: If two shapes are identical, we say they are
Similar shapes means the two shapes are
If you double the sides, does everything double?
Congruent
perfect enlargements of each other (corresponding sides are in the same proportion)
No the area is four times so stick to linear measure

23. Aims of the Session
To build understanding of mathematics and it’s development throughout KS2
To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods)
To enhance subject knowledge of the pedagogical approaches to teaching mathematics

24. Where now? By the next meeting, I am going to trial/action... *
Where next?
Progress in fractions, decimals and percentages