1. We didn’t do it like that when I was at school! A Whistle-stop Tour of Calculation at Sonning CE Primary School

2. NO YOU’RE RIGHT – YOU DIDN’T!

3. Aims of the Evening A bit of background It ain’t what you do … Addition and subtraction –Examples of informal and expanded methods Multiplication and division –Examples of informal and expanded methods What you can do to help your child

4. A bit of background … Until Year 4 there is a structured approach with emphasis on mental calculation, with annotations where appropriate. Only when mental calculations are secure are standard written methods introduced. Up to the end of Year 3, the emphasis mental methods, with calculations recorded in horizontal number sentences, and with some informal jottings for more challenging numbers. In Years 4 to 6, children are taught more formalised written methods of calculation, starting with expanded methods and working gradually towards more compact standard methods by the end of Year 6.

5. The overall aim is that when children leave primary school they: have a secure knowledge of number facts and a good understanding of the four operations; are able to use this knowledge and understanding to carry out calculations mentally and to apply general strategies when using one- digit and two-digit numbers and particular strategies to special cases involving bigger numbers; Primary Framework Calculation aims

6. make use of diagrams and informal notes to help record steps and part answers (JOTTINGS) when using mental methods that generate more information than can be kept in their heads; have an efficient, reliable, compact written method of calculation for each operation that children can apply with confidence when undertaking calculations that they cannot carry out mentally; use a calculator effectively, using their mental skills to monitor the process, check the steps involved and decide if the numbers displayed make sense.

7. Aim of calculations EFFICIENCY –USED APPROPRIATELY (NOT WHEN A MENTAL METHOD WOULD BE QUICKER) –CLEAR –QUICK –CORRECT! THE WAY TO ACHIEVE THIS MIGHT NOT ALWAYS BE WITH A STANDARD WRITTEN METHOD

8. It ain’t what you do … Importance of understanding not just doing Key - Place Value ?

9. What’s new pussycat? In teaching of the four operations - now a clearer link between early counting strategies and understanding of place value needed for more formal methods.

10. We want children to ask themselves: Can I do this in my head? Can I do this in my head using drawings or jottings? Do I need to use an expanded/compact written method? Do I need a calculator?

11. Addition and Subtraction Subtractions from the exercise book of John Hayne, 1715. Note that after the subtraction an addition is carried out as a check.

12. Addition and Subtraction Essential that they are taught as the INVERSE of one another Easy to demonstrate on a number line Later vital that children understand the commutative nature of addition (and multiplication) and non commutative nature of subtraction (and division) – for algebra 3 + 2 = 5 and 2 + 3 = 5 5 – 2 = 3 BUT 2 – 5 = 3

13. Addition Vocabulary add addition more/more than plus increase sum score total greater than altogether

14. Addition on The Empty Number Line Bridging the gap between counting and place value addition Home and Preschool Songs and games counting on/back in ones Mrs Hopper’s class Column addition with carried numbers Easy to record mental solution steps – could be alien to us!

15. Home and Preschool Songs and games counting on/back in ones Greater emphasis on mental maths before any written recording Children’s own recording with their own symbols or implicit ones Informal methods such as the empty number line for recording mental jumps 1 st stage of column method (partitioning adding 10s first – link with number line) Mrs Hopper’s class (Acorn) Column addition with carried numbers YEAR 4 - IF THEY ARE READY!

16. Addition on The Empty Number Line – Counting on (not crossing the tens boundary) More efficient

17. Addition on The Empty Number Line – Counting on =(crossing the tens boundary) More efficient

18. Over to you Have a go at the calculation 34 + 52 (counting on but NOT crossing tens boundary) Draw an empty number line Label left end with 34 (lowest number) Count up 52 in groups of 10 from 34 then add the ones/units We would encourage to children to estimate first so they can check the reasonableness of their answer

19. SUBTRACTION Inverse of addition Children find subtraction/learning subtraction facts much more difficult –Do they have enough experience in counting backwards? Taking away V Difference – from reception onwards. Utilises children’s generally stronger addition skills

20. Subtraction Vocabulary minus less than difference between How many more to make …? decrease by take away How many fewer is … than … ? How many more is... than...? How many less is... than...?

21. 16 - 9 1 Mistakes children make:

22. Decomposition (Taking away method)

23. Apparatus used to support more kinesthetic learners

24. Finding the difference (Counting up method)

25. Your turn Finding the difference Set out your calculation thus: Estimate The first question to ask is: “What do I need to add to 9 to make it up to the next whole ten?” Record this in the Units column. Record the number you have counted up to in brackets next to it Then ask: “What do I need to add to the number in brackets to make it up to the next hundred?” Record this in the tens column. Record the number you have counted up to in brackets next to it.

26. Then ask: “What do I need to add to the number in brackets to make it up to the TOTAL?” Record this in the hundreds column. Record the number you have counted up to in brackets next to it. Add the numbers (not those in brackets) together. This should be the difference between the two numbers.

27. Check with your estimate –does it seem right? Add your answer to the smaller number in the question. It should give you the number you took away from.

28. Multiplication and Division Understanding inverse essential Multiplication can be seen as repeated addition The compactness of traditional standard methods, and working with digits without referring to their values, hinders many children from understanding why a method works. Children who do not know their tables and division facts fail to make progress with calculations like 34 x 7 simply because they cannot recall multiplication table facts. Their effort to work out a simple fact like 4 x 7 diverts them from thinking about the process they are following, and they lose their way.

29. Multiplication Vocabulary lots of groups of times product multiplied by multiple of _____ times as big

30. Learning tables and division facts Start with 1, 2, 5, 10 and don’t forget 0! Learn tricks x 4 is x 2 x 2 (double and double again) Learning styles are important – songs, games, posters, flashcards Must learn division facts shortly after times table is learned Use commutative law when testing multiplications e.g 3 x 8 is the same as 8 x 3 Have to be learned out of order Only so much we can do in school – has to be commitment on part of the child. MUST BE PRACTICED REGULARLY!

31. From John Hayne's exercise book, 1719

32. Mistakes children make: 76 x 8 5648 67 x 54 268 335 603 101 r 5 7 847

33. Informal written multiplication – The Grid Method 12 024 24 x 6 Partition the numbers 20 4 6 =144

34. Informal written multiplication – The Grid Method 1800240 21028 34 x 67 30 4 60 7 2040 238+ 2278 For HTU x TU grid is inappropriate – too many numbers = more chance of an error

35. Division Division can be seen in two ways – repeated subtraction sharing Children generally find it the most difficult of the four operations -division facts are not learned alongside times tables facts -less flexibility in methods taught - too early introduction of standard methods -for repeated subtraction methods to work children have to be good at using recording to help them to keep track of their methods – they can’t!

36. Division Vocabulary shared by divided by shared equally between group into equal groups of divided into divisible by

37. Division as grouping

38. 8 6 48 ÷6= 6 42 48-6 36 42-6 6 30 36-6 6 24 30-6 6 18 24-6 6 6 12 18-6 6 6-6 0 12-6 6 How many 6’s have been subtracted? That means, there are 8 sixes in 48!

39. And as the numbers get bigger … …taking away one group at a time is not enough What’s the biggest group of _____ that you can take away from ________? This method relies on being able to multiply by 10 or doubling then multiplying by 10 etc And subtraction!

40. Three written division methods … … shamelessly stolen from someone with too much time on their hands!

41. Long Division Methods CHUNKING or MULTIPLES OF THE DIVISOR METHOD

42. We are going to try to solve 839 ÷ 27

43. 8 3 9 - 2 7 0(27 x 10 = 270)

44. 839 ÷ 27 8 3 9 - 2 7 0(27 x 10 = 270) 5 6 9

45. 839 ÷ 27 8 3 9 - 2 7 0(27 x 10 = 270) 5 6 9 2 9 9

46. 839 ÷ 27 8 3 9 - 2 7 0(27 x 10 = 270) 5 6 9 2 9 9 2 9

47. 839 ÷ 27 8 3 9 - 2 7 0(27 x 10 = 270) 5 6 9 2 9 9 2 9 - 2 7 (27 x 1 = 27) 2

48. 839 ÷ 27 8 3 9 - 2 7 0(27 x 10 = 270) 5 6 9 2 9 9 2 9 - 2 7 (27 x 1 = 27) 2 10 + 10 + 10 + 1 = 31

49. 839 ÷ 27 = 3 1r 2 Or 31 2 27

50. So how can you help your child? Help your child learn their addition/subtraction facts and times tables/division facts and impress upon them how important they are Ask your child to explain the method and if they are unsure –PLEASE don’t force them to use a method that you are confident in –DO talk about how you do maths generally –Wait until the Monday and encourage your child to go and ask or go with your child to have it explained

51. Encourage your child to estimate and check the reasonableness of their calculations Encourage them to use jottings to support their mental work but to set out standard written methods neatly Give tons of praise and encouragement DON’T EVER SAY, WITH PRIDE, THAT YOU WERE RUBBISH AT MATHS!