1. Arithmetic: Mental Calculation
PGCE
Arithmetic: Mental Calculation
Resources needed for this workshop –
Ross and his cubes video – on shared drive
Cubes / counters for demonstrations
Models and Images sheets for addition and Subtraction
Examples of arrays
Vocabulary charts
follow me cards
Numicom
Examples of jottings

2. Learning Objectives:
Be familiar with the models of the four operations
Be aware of the properties of the four operations
Be aware of the relevant vocabulary
Become familiar with a range of mental strategies
Consider the importance of structured jottings

3. What do we mean by addition, subtraction, multiplication and division?
SHOW ROSS AND HIS CUBES CLIP

4. Understanding addition, subtraction, multiplication and division
In order to be able calculate using the four operations a child needs to know:
The different models of addition, subtraction, multiplication and division
The properties of the four operations
Vocabulary

5. Models of addition Combining Counting on
1, 2, and , together makes , 2, 3, , 5
Counting on
and together makes , , 5

6. Models of subtraction Taking away Counting back Difference
Can you find an example for each?

8. The Singapore Bar Method Addition - Aggregation
There are 3 footballs in the red basket 2
footballs in the blue basket. How many
footballs are there altogether?

9. Addition - Augmentation
Peter has 3 marbles. Harry gives Peter 1 more marble. How many marbles does Peter have now?
Concrete Abstract

10. Subtraction - Comparison Model
Peter has 5 pencils and 3 erasers. How
many more pencils than erasers does he
have?

11. Moving to the abstract
Peter has 5 pencils and 3 erasers. How many more pencils than erasers does he have?

12. Generalisation

13. Giving meaning to calculations
Number stories for 5+3=8
Which models would you use?
I have 5 sweets and my friend gave me 3 more. How many do I have altogether?
My sister is 5 years old. How old will she be in 3 years time?

14. Number stories for 8-3=5
Which models would you use for children?
There are 8 apples on a tree. The squirrel ate 3. How many were left?
I am 8 and my sister is 3. How many years older am I than my sister?
I have 3 conkers and my friend has 8. How many more has he got than me?

15. Models for multiplication
What is multiplication?
Repeated addition
Lots of / groups of
Arrays
Scaling (n times as many, as long, as heavy…)

16. Models for division
What is division?
Sharing (equal)
Grouping, linked to:
Repeated subtraction

17. ITPs available to support
Arrays
Grouping & number line models

18. Vocabulary
Addition
Subtraction
Multiplication
Division

19. Properties
Write some number sentences using the numbers above.
Use all four operations
eg = 7

20. Property 1: Inverse
4 + 3 = 7
So …
7 – 3 = 4
3 + 4 = 7
So…
7 – 4 = 3
4 x 3 =12
So …
12 ÷ 3 = 4
3 x 4 = 12
12 ÷ 4 = 3

21. Property 2:Commutativity
Does it matter which way round the two operations are done?
4+3= =
Does it matter in which order you add these numbers together? =
Addition is commutative
Which of the other operations are commutative?

22. Property 3: Associativity
How would you do this, which pair would you start with? =
Does … (18+36)+4 = 18+(36+4)
Is addition associative?
Now try these, put the brackets in different places so that you start with different pairs.
12 – 7 - 2
2 x 3 x 4
12 ÷ 6 ÷ Which are associative?

23. Property 4: Distributive
7 x 13 = 7 x (10 + 3) = (7 x 10) +(7 x 3) 7
Is division?

24. Summary 1 Models, Links and Properties
Addition
Subtraction
Models
Links
Properties
● combining sets
● taking away from a set
● counting on or back
(number line)
● counting on (number line)
● difference between
INVERSES
Commutative
Neither
Associative

25. Summary 2 Models, Links and Properties
Multiplication
Division
Models
Links
Properties
● repeated addition
● repeated subtraction
● lots of / groups of
● sharing
● arrays
● groups of
● scaling
INVERSES
Commutative
Neither
Associative
Distributive over addition

26. EYFS 2013 and National Curriculum 2014
EYFS Early Learning Goal:
Use quantities of objects to add and subtract two single digit numbers
Count on and count back to find answers
National Curriculum 2014:
Solve problems using concrete objects, pictorial representations and arrays (year 1/2)
Use inverse relationships to check addition and subtraction calculations (year 2)

27. Cont….
Show that addition of two numbers can be done in any order (commutative) and subtraction cannot (year 2)
Show that multiplication of two numbers can be done in any order (commutative) and division cannot (year 2)
Estimate the answer and use inverse operations to check answer (year 3)
Solve problems with scaling (year 3)
Use commutativity in mental calculations (year 4)
Solve problems using distributive law (year 4)

28. Mental Strategies

29. Rover has left his bone on the other side of the road
Rover has left his bone on the other side of the road. He can only get there by treading on boxes with an answer that is 7 (number bonds)
3-1
6+2
6-2
1+1
5-1
5+7
0+7
9-2
1+6
1+3
4+2
5+2
6+6
2+5
3+4
8-1
2+3
4+3
6+1
4+4
4+0
8-3
7-6

30. Mental Strategy 1: Number Bonds
Not just number bonds for 10
Extend to number bonds for numbers up to 20
Make it visual for understanding
Useful apparatus:
Cuisenaire / Colour rods Numicom 8

31. Mental strategy 2: Partitioning
Use arrow cards to help children deconstruct numbers and combine multiples of hundred / ten / ones. (Later on introduce exchanging) 1 4 2 3
= ( ) + (3 + 1) Links to associativity

32. Mental strategy 3: Bridging through 10
To use this strategy children first need to know number bonds to 10 and partitioning.
Start visually using apparatus such as Nubicom / Colour Rods: 8 5 8 2 3
Children need to know number bond of 8 to make 10
5 is partitioned into 2 + 3
This is then extended to 2 digit numbers and the apparatus is replaced by empty number line jotting (covered later)

33. Mental strategy 4: adding / subtracting 9
29 + 9

34. Strategy 5: Complementary Addition
Or subtraction by addition.
200 – How could you model this?
In this calculation you are looking at the size of the gap between the two numbers.

35. Other Mental Strategies:KS1
counting on
using known facts e.g. doubles
derive facts e.g. near doubles
counting on/back in ones and tens
adding or subtracting 9 by adding or subtracting 10 and adjusting by 1
looking for number bonds to 10

36. Other Mental Strategies: KS2
partitioning numbers and dealing with the multiples of 10 first
adjusting numbers e.g. up or down to the nearest 10
using known facts to derive new facts
looking for number bonds of 10 or 100, especially when adding together more than two numbers
subtraction using complimentary addition and compensation methods

37. Try these….using mental strategies= = = = =
Discuss your strategies

38. Skills needed before introducing Multiplication and Division Mental Strategies
Working out 3x4 by counting out three groups/sets of four
Counting in equal jumps along the number line – ‘five, ten, fifteen, twenty’
Starting with tables for 1,2 and 10, knowing by heart facts such as ‘four tens’& progressing to facts in 5x table, then others
Recognising that multiplication can be done in any order – eg realising that 5x2 is the same as 2x5 – commutativity

39. Learning Multiplication Tables
Children need to understand what multiplication tables are.
5 x 3 is displayed in stamps
2.They need to understand the commutative law

40. Counting in 2s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100

41. Counting in 4s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100

42. Developing Multiplication and Division Mental strategies
1. Building on known facts:
If you know 3x5=15… what else do you know?

43. Developing strategies
2. Multiply and divide by multiples of 10 with whole and decimal numbers (link to place value) 3. Doubling and Halving

44. Have a go
509 x 3
5 x 23 x 20
112 ÷ 8
750 ÷ 25
Which strategies did you use?

45. Issues: Mental Calculations
methods are all valuable if they are quick and accurate
emphasis on place value and usually work from left to right
May involve adjusting numbers
Need a distinction drawn between special cases and general strategies – a repertoire from which you select according to the particular numbers
Need to be explained in words and described using correct equations (partly to check understanding)
May be accompanied by structured jottings

46. Structured Jottings Reflect and support mental strategies
May need to be tidied up for an audience
Should lead to shortened forms
Are flexible

47. Examples of children’s jottings
Children’s recording in Reception
Yr 1. My shepherd looks after 8 sheep but has lost 5 and he has 3 left..

48. Key Stage 1 examples: Children’s recording in Year 2
1. Write the answer: =
Taken from Standards at Key Stage 1, QCA 2001 p.37
2. Write the number that is half of 38
30 + 8
15 + 4
As above, p 39

49. Open number lines 33 2 23 5 38 40 63 25 30 63
Answer 63
246 – 78
using complementary addition and looking at the difference 20 2
146 78 80
100
246
Answer = 168

50. Ofsted 2011
Stresses the importance of children demonstrating a fluency in calculating, solving problems and reasoning about number.
Key findings:
Practical, hands-on experiences are crucial in EYFS and KS1 couple with opportunities to develop mathematical language.
Understanding place value, fluency in mental methods and good recall of number facts.
Subtraction should be taught with its inverse addition and division taught alongside its inverse, multiplication.

51. Ofsted 2011 (cont.)
Children need to increasingly develop more sophisticated mental and written methods
Children need to be taught to be flexible in their thinking and approaches
Needs to be a strong emphasis on problem solving
Teachers need to recognise and quickly intervene when misconceptions occur so that progress is not impeded
Teachers need good subject knowledge and subject specific teaching skills.
Ofsted (2011) Good practice in primary mathematics Manchester: Ofsted (Full report available from:

52. Children who experience problems
do not look for alternative methods
overlook number properties
try to replicate standard written methods in their heads
depend on counting strategies
have limited strategic methods
do not treat number holistically

53. Useful resources
NCETM calculations microsite and videos to support teaching of calculations
Teaching Mental Calculation booklet in your Arithmetic self study work book
Resources on Moodle linked to mental calculation workshops.