PGCE
Arithmetic: Mental Calculation

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

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

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

Models of addition
• Combining
1, 2, 3 and 1, 2 together makes 1, 2, 3, 4, 5
• Counting on
3 and 2 together makes 3, 4, 5

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

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?

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

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

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

Generalisation

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?

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?

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

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

• Arrays
• Grouping & number line models

• Addition • Multiplication
• Subtraction • Division

3 4 7 12
Write some number sentences using the numbers above.
Use all four operations eg 3 + 4 = 7

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

Property 2:Commutativity
Does it matter which way round the two operations are done?
4+3= 3+4=
Does it matter in which order you add these numbers together?
15 + 7 + 5 + 3 =
Addition is commutative
Which of the other operations are commutative?

Property 3: Associativity
How would you do this, which pair would you start with?
• 18+36+4=
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 ÷ 3 Which are associative?

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

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

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

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)

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)

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
5+7
4+2
3+4
1+1
6+2
0+7
5+2
8-1
4+4
6-2
9-2
6+6
2+3
4+0
1+1
1+6
2+5
4+3
8-3
5-1
1+3
6+2
6+1
7-6

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
8
Numicom
6 + 2 https://www.ncetm.org.uk/resources/40533
3 + 4 7

Mental strategy 2: Partitioning
Use arrow cards to help children deconstruct numbers and combine multiples of hundred / ten / ones. (Later on introduce exchanging)
4
3
2 1
43 + 21 = (40 + 20) + (3 + 1) Links to associativity https://www.ncetm.org.uk/resources/40534

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
8
5
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)

Mental strategy 4: adding / subtracting 9
29 + 9

Strategy 5: Complementary Addition
Or subtraction by addition.
200 – 56 How could you model this?
In this calculation you are looking at the size of the gap between the two numbers.
4 40 100
56 60 100 200

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

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

Try these….using mental strategies
• 47 + 32 =
• 39 + 18 =
• 401 + 395 =
• 57 - 29 =
• 1000 - 989 =
Discuss your strategies

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

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

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 10
0

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

1. Building on known facts:
If you know 3x5=15… what else do you know?

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

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

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

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

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..

Key Stage 1 examples: Children’s recording in Year 2
1.
Write the answer: 51- 27
=
24 3 + 20 + 1
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

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

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.

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: www.ofsted.gov.uk/resources/110140)

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

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.