Arithmetic: Mental CalculationArithmetic: Mental Calculation

PGCE

Learning Objectives: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

SHOW ROSS AND HIS CUBES CLIP

What do we mean by addition, subtraction, multiplication and division?What do we mean by addition, subtraction, multiplication and division?

Understanding addition, subtraction, multiplication and divisionUnderstanding 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 additionModels 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 subtractionModels of subtraction• Taking away

• Counting back

• Difference

Can you find an example for each?

The Singapore Bar MethodAddition - AggregationThe Singapore Bar MethodAddition - Aggregation

There are 3 footballs in the red basket 2

footballs in the blue basket. How many

footballs are there altogether?

Addition - AugmentationAddition - Augmentation• Peter has 3 marbles. Harry gives Peter 1

more marble. How many marbles does Peter have now?

Concrete Abstract

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

Moving to the abstractMoving to the abstract• Peter has 5 pencils and 3 erasers. How

many more pencils than erasers does he have?

GeneralisationGeneralisation

Giving meaning to calculationsGiving meaning to calculationsNumber 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 multiplicationModels for multiplication

What is multiplication?

• Repeated addition

• Lots of / groups of

• Arrays

• Scaling (n times as many, as long, as heavy…)

Models for divisionModels for division

What is division?

• Sharing (equal)

• Grouping, linked to:

• Repeated subtraction

ITPs available to support ITPs available to support

• Arrays• Grouping & number line models

Vocabulary Vocabulary• Addition

• Subtraction

• Multiplication

• Division

3 4 7 12

Write some number sentences using the numbers above.

Use all four operations

eg 3 + 4 = 7

PropertiesProperties

Property 1: Inverse Property 1: Inverse 4 + 3 = 7

So …

7 – 3 = 4

3 + 4 = 7

So…

7 – 4 = 3

4 x 3 =12So …12 ÷ 3 = 4

3 x 4 = 12So …12 ÷ 4 = 3

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 2:Commutativity

Property 3: Associativity 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: DistributiveProperty 4: Distributive

7 x 13 = 7 x (10 + 3) = (7 x 10) +(7 x 3) 10 3

7

Is division?

Summary 1Models, Links and Properties

Summary 1Models, Links and Properties

Addition Subtraction

Models

Links

Properties

● combining sets ● taking away from a set● counting on or back (number line)● difference between

NeitherAssociative

Commutative

● counting on (number line)

INVERSES

Summary 2Models, Links and Properties

Summary 2Models, Links and Properties

Multiplication Division

Models

Links

Properties

● repeated addition ● repeated subtraction● sharing

● groups of

Neither

Associative

Commutative

● lots of / groups of

INVERSES

● arrays● scaling

Distributive over addition

EYFS 2013 and National Curriculum 2014EYFS 2013 and National Curriculum 2014EYFS 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….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)

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 6+2

3+4 8-1 2+3 4+3 6+1

1+1 4+4 4+0 8-3 7-6

Mental Strategy 1: Number BondsMental 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

6 + 2

https://www.ncetm.org.uk/resources/405333 + 4 7

Mental strategy 2: PartitioningMental strategy 2: Partitioning

Use arrow cards to help children deconstruct numbers and combine multiples of hundred / ten / ones. (Later on introduce exchanging)

4 2 3

1

43 + 21 = (40 + 20) + (3 + 1) Links to associativity

https://www.ncetm.org.uk/resources/40534

Mental strategy 3: Bridging through 10Mental strategy 3: Bridging through 10To 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

5 is partitioned into 2 + 3Children need to know number bond of 8 to make 10

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 9Mental strategy 4: adding / subtracting 9

29 + 9

Strategy 5: Complementary AdditionStrategy 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:KS1Other 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: KS2Other 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 strategiesTry 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 StrategiesSkills 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 TablesLearning Multiplication Tables1. Children need to understand what

multiplication tables are.

2.They need to understand the commutative law

5 x 3 is displayed in stamps

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

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

Counting in 4s

Developing Multiplication and Division Mental strategies Developing Multiplication and Division Mental strategies

1. Building on known facts:

If you know 3x5=15… what else do you know?

Developing strategies 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 goHave a go

• 509 x 3

• 5 x 23 x 20

• 112 ÷ 8

• 750 ÷ 25

Which strategies did you use?

Issues: Mental CalculationsIssues: 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 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 jottingsExamples of children’s jottingsChildren’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 Key Stage 1 examples: Children’s recording in Year 2

1. Write the answer: 51- 27 = 24 3 + 20 + 1Taken from Standards at Key Stage 1, QCA 2001 p.37

2. Write the number that is half of 38

30 + 8 15 + 4As above, p 39

Open number linesOpen number lines25 + 38

25 30 63

5 33

38 40 63

2 23

246 – 78

78 80 100 246

2 20 146using complementary addition and looking at the difference

Answer 63

Answer 146 + 20 + 2 = 168

Ofsted 2011Ofsted 2011Stresses 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.)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 problemsChildren 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 resourcesUseful 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.