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. Subtraction
Make up a word problem on a slip of paper that represents the calculation
7 – 2 = 5

3. Subtraction problems
I had 9 apples but my rabbit ate 3 of them. How many did I have left?
I had 9 apples. My friend Harry had 3 apples. How many more apples did I have?
Different models of subtraction:
The first is take away – Partitioning
The second is difference – Comparison (where they need to draw 12 counters!)
The third is missing number (difference) – Inverse of Addition
Subtraction is the family – take away alone over emphasises the cardinal and inhibits understanding
Haylock has five models in total but don’t want to go into such detail here

4. The Bar Model – How does it support understanding?3 ? 9
The Bar Model – How does it support understanding?

5. Models for subtraction
Counting back on a number line
Also:
Knowledge of place value and number bonds can support more efficient calculating
- 5
Issue:
Number line helps to stop ‘counting all’. 12 7 10 -3 -2
Finding the difference on a number line
Issue:
Useful when two numbers are ‘close together’, use of number bonds and place value can help. 7 5 10 12 5 2

6. Subtraction
Imran has 43 conkers; he gives 24 away to his friends. How many does he have left?
43 – 24 = 43 -1 19 20 -3 23
-10 33
-10
(Animated slide)
This slide demonstrates to parents the process of using an empty number line to support children’s thinking in subtraction.
It might well be how they (the parents) would work this mentally using an imaginary number line.
Children will experience this type of recording from Y2. The jumps are below the line to show ‘counting back’.
19 conkers

7. Using a numberline
Sam has saved 93p, Amy has 55p. How much more money does Sam have than Amy?
93 – 55 = 93 +3 60 +5 90
+30 55
(Animated slide)
Another subtraction calculation, this time involving difference.
Here the child has counted on from 55 to 93, recording the jumps on a number line. Having a visual image like this supports the child with what is going on in their head. The big jump of 30 may initially be made with three jumps of 10.
38p more

8. Robber maths
43 – 13
The number you need to subtract is small enough to “pick it up and take it away”
Explain strategy of Robber Maths
Taking away because the amount is small enough to manipulate, remember and “carry away”

9. Mind the gap
The gap between the two numbers is smaller so it is more efficient to find the difference (probably by counting on)
Counting on because the gap is handleable and easy to manipulate – children can be taught to explicitly think whether each strategy is more appropriate
Some calculations do not fit either strategy and this is where jottings can come in.

10. Robber maths? – Mind the gap?
101 – 99
63 – 21
84 – 78
1006 – 999
86 – 14

11. Starting Point
Before launching in to the expectations of KS2, the following are the new National Curriculum expectations for year 2:

12. Building the Journey
Year 3 Addition and Subtraction upto 3 digits using formal methods
Year 4 Addition and Subtraction upto 4 digits using formal methods
(Solve simple measure and money problems involving decimals to 2 dp)
Year 5 Addition and Subtraction more than 4 digits using formal methods
(Solve problems involving number up to 3 dp)
(They mentally add and subtract tenths, and one-digit whole numbers and tenths)
(They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, = 1)).
All years groups also refer to "estimation", "inverse operations" to check, "problem solving"

13. Trading Game H T U 4 30
Groups need place value base board, a die, a bank of base 10 materials (dienes), sets of 0-9 cards. They start with 50 on the board in 10s (rods). (Alternatively they can play it with 10ps and 1ps – another visual model)
Children take turns to roll the dice and take that number away. The children will first need to exchange a ten into units to take their number away.The children put their ‘tens’ and ‘ones’ on the place value board with the correct digit card underneath to record the remaining total. The winner is either the person with the most accumulated once the 50 has gone or (if playing with parallel starting amounts) the first person to take all their 50 away, having taken it in turns to throw the dice.
Then demo on the flip chart the expanded method of decomposition eg.
227 – 134
Show the moving of the 100 into the middle column and point out how this relates to the decomposition on the place value mat in the trading game.
Show other examples if you need to – this expanded method is a vital step for children who have misconceptions
Show some of the misconceptions in decomposition – don’t move on too fast. Move to expanded method of subtraction. Model with the Dienes
43 – 27. Exchanging a 10 –dienes first then expanded written methods.
Then demonstrate decomposition method 426 – 289.
Go back to flipchart with problems and recap.

14. 1 10 10 10 10 10 6 7 2 10 1 10 10 1 1 - 7 4

15. 1 1 1 10 10 1 1 10 10 1 6 7 2 10 1 10 - 7 4

16. 1 1 1 10 10 1 1 10 10 6 7 2 10 1 1 10 - 7 4

17. 1 1 1 10 10 1 1 10 10 6 7 2 10 1 10 1 - 7 4

18. 1 1 1 10 10 1 1 10 10 6 7 2 10 1 10 - 7 4 1

19. 1 1 1 10 10 1 1 10 6 7 2 1 - 7 4 1 5

20. 1 1 1 10 10 1 1 10 6 7 2 1 - 7 4 1 5

21. 1 1 1 10 10 1 1 6 7 2 1 10 - 7 4 1 5

22. 1 1 1 10 10 1 1 6 7 2 1 - 7 4 10 1 5

23. 1 1 1 10 10 1 1 6 7 2 1 - 7 4 1 2 5 10

24. 1 1 1 10 10 1 1 5 2 1 + 4 7 10

25. 1 1 1 10 10 1 1 5 2 1 + 4 7 10

26. 1 1 1 10 10 1 1 1 5 2 + 4 7 10

27. 1 1 1 10 10 1 1 1 5 2 + 4 7 10

28. 1 1 1 10 10 1 1 1 1 1 1 1 1 1 5 2 + 4 7 10 12

29. 1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 5 2 + 4 7 10 2 1

30. 1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 5 2 10 + 4 7 2 1

31. 1 1 1 10 10 1 1 1 1 1 1 1 10 1 1 10 5 2 + 4 7 2 1

32. 1 1 1 10 1 10 1 1 1 1 1 10 1 10 1 1 5 2 + 4 7 2 1

33. 1 1 1 10 10 1 1 10 1 1 1 1 1 10 1 1 5 2 + 4 7
Show Video of introducing written column subtraction 7 2 1

34. Year 3
Essentially year 3 becomes a time when more "formal methods are introduced". This needs to take place as/when the individual pupil is ready:
e.g. U T U T 10 1 10 1 10 1 10 1 10 1 10 10 1 10 10 1 10 1 1

35. Subtraction with 3-digits (an end of Year Objective)
e.g. U T H U T H U T H
100 10 1
100
100 10 1
100 10 1
100
100 10 1
100 1
100
100 10 1
100 10 1
100 1

36. "Estimating Answers & Checking Using Inverse Operations"
e.g U T H U T H

37. Some day-to-day pointers:
If some pupils need refreshing (say on single digit addition or partitioning) how will you be using your TA effectively? Is your TA going to be up to speed on the different strategies that you will be employing?
How/when is students' work marked and assessed? Can you reduce your workload here and at the same time give better feedback to pupils?
If a pupil has "mastered" the process in the Autumn term, they need their knowledge extended in the spring/summer terms. Often problem solving is the key here.
How to we ensure that work doesn't fall into the "boring" category? Initially pupils will be motivated by ticks on a page, but if the style of question become repetitive, there might be a tendency to "switch off"

38. Careful not to write jibberish!
Spicing up Addition and Subtraction:
Arithmagons
Number Walls
Careful not to write jibberish!
Some of these will need real resilience, but the sense of achievement will be much greater once solved!

39. Magic Squares
Ink Blots/
Missing Digit
Darts?! T U 2 + 3 5
Cryptarithms/ Alphametics T U 7 5 + 3 1 H

40. Year 4
The strategies met in year 3, extend into year 4 - with addition and subtraction now with four digits. My suggestion would be:
Although pupils might be able to naturally extend the method, revisit the kinaesthetic examples so they link their new objective to prior learning. If some pupils need longer working kinaesthetically than others - fine!
Some very visual learners will even remember "counter colours" from the previous year, so ensure complete consistency between year groups.

41. Up to Four-Digit SubtractionH Th
1000
Can you scaffold a hierarchical set of questions for subtraction?
Is the column approach always sensible?
100 10 1

42. More than Four-Digit Subtraction
Year 5 U T H
More than Four-Digit Subtraction U T H Th
10Th
10000
1000
Can you scaffold a hierarchical set of questions for subtraction?
Is the column approach always sensible?
100 10 1

43. Note - all the "livening up" and "problem solving" skills from year 3 should also be embraced in years 4 & 5. As too should estimation and inverse operations to check.

44. 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

45. Where now? By the next meeting, I am going to trial/action... *
Where next?
Progress in multiplication
Progress in division and ratio.
Progress in fractions