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Progression in Calculations

December 2014

version 1.3

Billingshurst Primary School Progression in Calculations CONTENTS page 3 Introduction Pages 4 & 5 2014 National Curriculum statutory requirements for calculations page 6 Place Value page 7 The Importance of Mathematics in Early Experiences pages 9 - 25 Addition (contents for this topic on page 8) pages 27 - 41 Subtraction (contents for this topic on page 26) pages 43 - 59 Multiplication (contents for this topic on page 42) pages 61 - 75 Division (contents for this topic on page 60)

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Billingshurst Primary School Progression in Calculations Introduction

What is the purpose of this document?

This document has been designed for parents, carers, children , governors and teachers at Billingshurst Primary School as a guide to the progression for calculations in addition, subtraction, multiplication and division. The contents are not an exhaustive list of all possible methods, rather the progression followed through the year groups at Billingshurst Primary School. The children need to have consistent and accurate methods for solving problems by calculating and these are contained in this document. However, it should be noted that if children have an alternative method which they can use successfully, they will not be dissuaded from using it. Finally, you will notice that methods have a suggested year group. These are derived from the 2014 National Curriculum, which states expectations for each year group relating to methods and complexity.

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Billingshurst Primary School Progression in Calculations Place Value

Underpinning ideas

Numbers are made using the digits from 0 to 9 in the same way as words are made of letters from a to z. The value of a number depends on where the digits are placed, just as the meaning of a word depends on the order of the letters. For example the letters p, s, t and o can make: • post, stop, pots, tops

Similarly we can make lots of different numbers using the digits 4, 6, 3 and 9. In each example we will examine the value of the digit 4: 4639 (four thousand) 6439 (four hundred) 3964 (four units) 93.46 (four tenths) 4.936 (four units) and so on.

What place value looks like

Millions Hundreds

of Thousands

Tens of Thousands

Thousands Hundreds Tens Units Decimal

point tenths hundredths thousandths

M 100Th 10Th Th H T U t h th

1000000 100000 10000 1000 100 10 1 0.1 0.01 0.001

1/10 1/100 1/1000

Billingshurst Primary School Progression in Calculations Early experience in Mathematics

Why is this important?

Mathematics is all around us. Whether in a Mathematics lesson, popping to the shops, doing a spot of baking, totting up the accounts or filling the car with petrol. Often the way we experience Mathematics is informal, seldom do we use the scales at the supermarket to check we have enough broccoli to the nearest 10 grams for example. Given that Mathematics is experienced in ‘real life’ in so many different ways, the children need to be flexible in their approaches to it, and the best foundations are laid early. Singing songs involving counting is a great way to reinforce basic number skills, whether it be counting on (‘This Old Man’)’ or counting back (‘Five Little Ducks’). Here are some more examples: • 5 little speckled frogs • 1,2,3,4,5 once I caught a fish alive • 5 little monkeys jumping on the bed • 10 in a bed • 3 little men in a flying saucer • 5 currant buns in a baker’s shop As we shall see later, playing games such as Snakes and Ladders can also be beneficial in teaching basic concepts. Cookery is also a great activity that teaches a wide range of skills, from measuring to fractions and even being responsible for budgeting for the cost of the ingredients. These experiences will help support children to achieve elements of two of the Early Learning Goals which the children are expected to be working within at the end of Reception. Numbers: Children count reliably with numbers from one to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing. Shape, Space and Measure: Children use everyday language to talk about size, weight, capacity, position, distance, time and money to compare quantities and objects and to solve problems. They recognise, create and describe patterns. They explore characteristics of everyday objects and shapes and use mathematical language to describe them. 7

Billingshurst Primary School Progression in Calculations ADDITION Contents pages 9 – 11 Counting skills pages 12 – 16 Early addition page 17 Recording calculations pages 18 - 20 Adding ones and tens page 21 Partitioning page 22 Column addition page 23 Column addition with carrying page 24 Column addition with carrying – larger numbers page 25 Column addition with decimals

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Billingshurst Primary School Progression in Calculations Counting skills

The Language of Addition Methods

+ Add Addition More More than Plus Make Sum Total Altogether The same as Score One more, two more… ten more How many more to make…? How many more is… than…? How much more is…? How can we find out how many we’ve got?

Modelling 1:1 accurate counting, for example:

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“Can you touch each one as you say each number?” This is also known as 1:1 correspondence.

Numicon Base 10 Objects

Billingshurst Primary School Progression in Calculations Counting skills



Modelling systematic counting methods, for example:

10

How many altogether? Can you add 1 more ? How many will there be then? Can you match the number to the group of objects?

Numicon

Objects

Billingshurst Primary School Progression in Calculations Counting skills – conservation of number

Underpinning ideas Method

The children will use their understanding of the number system to develop ways of recording calculations using pictures.

For example:

The farmer had 9 cows. He places 3 cows in one field and the other 6 cows in a different field. The farmer still has 9 cows.

It is important for the children to understand that there are still 9 cows overall. This idea is called conservation of number – there are 9 cows no matter how the group is structured.

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Billingshurst Primary School Progression in Calculations Early addition



Combining two sets, for example:

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Can you put these 2 groups together? How many will there be in total?

Can you add on 1 more? How many will there be?

Numicon

Cuisenaire and number track

Objects

Billingshurst Primary School Progression in Calculations Early addition



Practical activities

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There is so much maths in a game of snakes and ladders! (1:1 correspondence, counting on, counting back, more than,

less than, addition and subtraction, numeral recognition, sequence and position of numbers.)

Billingshurst Primary School Progression in Calculations Early addition continued



Using resources:

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In this example, the resources are used to calculate 28 + 6 = 34

Numicon

Bead string





Using resources, such as a number line:

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In this example, the number line is used to calculate 28 + 6 = 34

This is achieved by finding 28 on the number line then counting on six.




Using resources, such as a 100 square:

16

In this example, the 100 square is used to calculate 28 + 6 = 34

Billingshurst Primary School Progression in Calculations Starting addition – RECORDING CALCULATIONS

Underpinning ideas Written method

The children will use their understanding of the number system to develop ways of recording calculations using diagrams.

For example, 4 + 3 = 7

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Numicon

Objects

Number sentence

Billingshurst Primary School Progression in Calculations Next steps in addition – ADDING ONES AND TENS

The progression of adding ones and tens

Counting on in steps of one: Counting on in steps of ten and then steps of one:

0 10 1 2 3 4 5 6 7 8 9

+1 +1 +1

4 + 3 = 7

Numicon


18



Counting on in steps of ten and then steps of one: The next step is to count on in steps of ten and then combine the ones in to a single step, in this case:

19

44 54 64 65 66 67

+1 +1 +1

44 + 23 = 67

+10 +10

44 54 64 65 66 67

+3

44 + 23 = 67

+10 +10 Cuisenaire and number track



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Now combine the tens into one step, and the ones into a second step: Bridging through ten – this example shows how children can use counting on in steps of ten and then different groups of ones:

44 54 64 65 66 67

+3

44 + 23 = 67

+20

27 37 40 42

27 + 15 = 42

+10 +3 +2

Billingshurst Primary School Progression in Calculations Addition – PARTITIONING


As the children move away from using number lines to support their calculations, they can still use the same theory to help find the answer. Partitioning means that the children identify how a number is made up by considering the place value columns and the digits in each one.

For example: 56 + 27 T U 56 can also be thought of as 5 6 or 5 Tens and 6 Units or 50 and 6. This can be proven by adding 50 to 6 to equal 56. T U 27 can also be thought of as 2 7 or 2 Tens and 7 Units or 20 and 7. Using this knowledge the children can begin to solve addition problems in a more abstract way: 56 + 27 = 83 As when using the number line, the Tens are added first and then the Units. (50 + 20) + (6 + 7) = 70 + 13 = 83

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Billingshurst Primary School Progression in Calculations Addition – COLUMN ADDITION


All the methods used so far can be applied to progressively larger numbers. The next evolutionary step in addition is column addition. This is based on the methods learnt to this point but the format is vertical rather than horizontal.

For example: 15 + 32 The numbers are set out in place value columns as shown below. This is very important as it ensures that Units are added to Units, Tens are added to Tens and so on. If the sum is laid out inaccurately it can lead to errors with Units being added to Tens for example. Each column is then calculated, always starting with the lowest place value column, in the case the Units column. You will notice that the largest number has been placed in the first row of the calculation. This is convention and occurs for the reasons explained in the ‘Calculation Order’ section.

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T U 3 2

+ 1 5

4 7

2 Units + 5 Units = 7 Units

3 Tens + 1 Ten = 4 Tens

Billingshurst Primary School Progression in Calculations Addition – COLUMN ADDITION WITH CARRYING



For example: 26 + 19. The sum is set out neatly in the appropriate place value columns. Using the same format as before, we can see that 6 Units plus 9 Units equals 15 Units. In mathematics we can only put one digit in each place value column when forming a number, so we cannot put 15 in the Units column. However, 15 Units would more commonly be written as 15, or 1 Ten and 5 Units. Therefore a 5 is placed in the Units column and 1 Ten is carried in to the Tens column (show in red). Then we add 1 Ten that as been carried and 2Tens and 1 Ten to give 4 Tens.

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T U

2 6 + 1 9

6 Units + 9 Units = 15 Units

T U 1

2 6 + 1 9

4 5

Billingshurst Primary School Progression in Calculations Addition – COLUMN ADDITION WITH CARRYING – LARGER NUMBERS



Carrying continues to work for larger numbers: 349 + 273

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H T U 1 1 3 4 9

+ 2 7 3

6 2 2

Billingshurst Primary School Progression in Calculations Addition – COLUMN ADDITION WITH DECIMALS



Column addition works equally well as a method when decimal places are involved in a calculation, as long as the sum is still set out neatly and accurately. The decimal point column can essentially be ignored and the sum calculated as usual.

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T U t h

8

5

6

3

9

2

1 2 2

1 1

1

+

Billingshurst Primary School Progression in Calculations SUBTRACTION Contents pages 27 – 29 Early subtraction page 30 Subtracting ones and tens page 31 Counting on page 32 Counting on with larger numbers page 33 Partitioning page 34 Expanded column subtraction page 35 Column subtraction page 36 Column subtraction with larger numbers page 37 Regrouping pages 38 - 41 Column subtraction with regrouping

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Billingshurst Primary School Progression in Calculations Early subtraction

The Language of Subtraction Methods

- Subtract Take (away) Less Minus Leave Bigger Smaller Difference How many are left/left over? How many have gone? One less, two less, ten less… How many fewer is… than…? How much less is…? Difference between

Practical games and songs, for example:

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The popular song ‘Five Little Speckled Frogs’.

At the end of each verse, one more ‘frog’ jumps in to

a pool. Children are counting back in ones.

Five little speckled frogs, Sat on a speckled log, Eating some most delicious bugs, Yum! Yum! One jumped in to the pool, Where it was nice and cool, Then there were four green speckled frogs, Glug! Glug! Four little speckled frogs, Sat on a speckled log, Eating some most delicious bugs, Yum! Yum! One jumped in to the pool, Where it was nice and cool, Then there were three green speckled frogs, Glug! Glug! And so on, until... One little speckled frog, Sat on a speckled log, Eating some most delicious bugs, Yum! Yum! Shejumped in to the pool, Where it was nice and cool, Then there were no green speckled frogs, Glug! Glug!

Billingshurst Primary School Progression in Calculations Early subtraction continued



Using resources, such as a bead string:

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In this example, the bead string is used to calculate 12 - 4 = 8

Billingshurst Primary School Progression in Calculations Early subtraction continued



Using resources, such as a 100 square or number line:

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In this example, the resources are used to calculate 12 - 4 = 8

Billingshurst Primary School Progression in Calculations Next steps in subtraction – SUBTRACTING ONES AND TENS


Counting back in steps of one: Counting back in steps of ten and then steps of one: The next step is to count back in steps of ten and then combine the ones in to a single step, in this case:

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Now combine the tens in to one step, and the ones in to a second step: Bridging through ten – this example shows how to children can use counting back in steps of ten and then different groups of ones:

0 10 1 2 3 4 5 6 7 8 9

-1 -1 -1

8 - 3 = 5

57 67 44 45 46 47

-1 -1 -1

67 - 23 = 44

-10 -10

57 67 44 47

-3

67 - 23 = 44

-10 -10

67 44 47

-3

67 - 23 = 44

-20

27 30 32 42

42 - 15 = 27

-10 -3 -2

Bead strings can also be used to support these calculations.

Billingshurst Primary School Progression in Calculations Subtraction – COUNTING ON


Subtraction is the inverse of addition – taking away rather than adding on. However, addition can be used to solve subtraction problems. This is because when subtracting we are finding the difference between two numbers. For example, the difference between 15 and 26 is 11, whether we subtract the two numbers or start at 15 and count up to 26. With a good understanding of what subtraction is, children may decide to solve problems by counting up.

The method of counting up from the smaller to the larger number can be recorded using number lines. The two numbers from the subtraction number sentence are placed at each end of the number line. 73 – 26 = 47 The process can be simplified by combining steps.

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26 73 30 70

+40 +3 +4

26 73 30

+43 +4

Billingshurst Primary School Progression in Calculations Subtraction – COUNTING ON WITH LARGER NUMBERS


Subtraction is the inverse of addition – taking away rather than adding on. However, addition can be used to solve subtraction problems. This is because when subtracting we are finding the difference between two numbers. For example, the difference between 15 and 26 is 11, whether we subtract the two numbers or start at 15 and count up to 26. With a good understanding of what subtraction is, children may decide to solve problems by counting up.

225 – 58 = 167 Or:

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60 220 100 200

+100 +20 +40 +2

58 225

+5

100

+125 +42

225 58

Billingshurst Primary School Progression in Calculations Subtraction – PARTITIONING


As the children move away from using number lines to support their calculations, they can still use the same theory to help find the answer. Partitioning means that the children identify how a number is made up by considering the place value columns and the digits in each one.

For example: 74 – 27 For 74 – 27 this involves partitioning the 27 into 20 and 7, and then subtracting from 74 the 20 and the 7 in turn. 74 – 27 74 – 20 = 54 54 – 7 = 47

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Billingshurst Primary School Progression in Calculations Subtraction – EXPANDED COLUMN SUBTRACTION


All the methods used so far can be applied to progressively larger numbers. The next evolutionary step in subtraction is column subtraction. This is based on the methods learnt to this point but the format is vertical rather than horizontal.

For example: 63 - 41 The numbers are partitioned then written under one another to be subtracted. The smallest place value column is the first to be subtracted, in this case the Units column.

63 – 41 = 22

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60

40

+ 3

+ 1 -

20 + 2

3 Units - 2 Units = 1 Unit

6 Tens - 4 Ten = 2 Tens

Billingshurst Primary School Progression in Calculations Subtraction – COLUMN SUBTRACTION



For example: 63 - 41 Column subtraction presents the same calculation but without partitioning first

63 – 41 = 22

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6

4

3

1 -

2 2

3 Units - 2 Units = 1 Unit

6 Tens - 4 Ten = 2 Tens

T U

Billingshurst Primary School Progression in Calculations Subtraction – COLUMN SUBTRACTION WITH LARGER NUMBERS



For example: 463 - 251

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Expanded method

400 + 60 + 3 - 200 + 50 + 1

200 + 10 + 2

Leading to

463 - 251

212

Billingshurst Primary School Progression in Calculations Subtraction – REGROUPING



For example: 84 - 37 The numbers are partitioned then written under one another to be subtracted, Units first and then Tens.

However, in this context it is not possible to take 7 away from 4. Of course there is an answer to 4 – 7, it equals-3, but column subtraction does not support negative numbers in the answer row. Therefore we need to regroup. NB – this used to be referred to as ‘borrowing’ but if we borrow something we would usually expect it to be returned, which does not happen in this case. This process is now described as regrouping. 37

80

30

+ 4

+ 7 -

In order to subtract 7 in the units column the unit above it needs to be larger than 7. We have already partitioned 84 as 80 +4 but it is also true that 84 could be partitioned as 70 + 14 as this still equals 84. The calculation can the be carried out.

80

30

+ 4

+ 7 -

80

30

+ 4

+ 7 -

70 14

+ 7 40

Billingshurst Primary School Progression in Calculations Subtraction – COLUMN SUBTRACTION WITH REGROUPING



For example:

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Regrouping is now shown in the place value columns, so 8 Tens becomes 7 Tens and 4 Units becomes 14 Units.

80

30

+ 4

+ 7 -

70 14

+ 7 40

80

30

+ 4

+ 7 -

8

3

4

7 -

4 7

7 14




Here an example shows regrouping from the Hundreds column to the Tens column. 400 + 30 can be partitioned into 300 + 130. The Tens column subtraction then becomes 130 minus 50 which equals 80.

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400 + 30 + 2

- 200 + 50 + 1

100 + 80 + 1

432

- 251

181

300 130 400 + 30 + 2

- 200 + 50 + 1

3 13




Here an example shows regrouping from the Hundreds column to the Tens column. Here both the tens and the ones digits to be subtracted are bigger than both the tens and the ones digits you are subtracting from. 30 + 2 is partitioned into 20 + 12, and then 400 + 30 can be partitioned into 300 + 130.

40

400 + 30 + 2

- 200 + 50 + 7

100 + 70 + 5

432

- 257

175

300 20 400 + 30 + 2

- 200 + 50 + 7

3 12 12

120

12




Here an example shows regrouping when dealing with a zero in a place value column. Here 0 acts as a place holder for the tens. Regrouping has to be done in two stages. First the 400 + 0 is partitioned into 300 + 100 and then the 100 + 2 is partitioned into 90 + 12.

41

400 + 0 + 2

- 200 + 50 + 7

100 + 40 + 5

402

- 257

145

300 100 400 + 0 + 2

- 200 + 50 + 7

3 9 12

90

12

Billingshurst Primary School Progression in Calculations MULTIPLICATION Contents pages 43 – 47 Early multiplication page 48 Repeated addition page 49 Commutative law of multiplication page 50 Arrays pages 51 & 52 Partitioning page 53 Multiplying by 10, 100 and 1000 pages 54 & 55 Grid method pages 56 & 57 Grid method with decimals page 58 Long multiplication page 59 Short multiplication

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Billingshurst Primary School Progression in Calculations Early multiplication

The Language of Multiplication Methods

X Lots of Groups of Times Multiply Once, twice, three times… ten times… ....times as big, long, wide… and so on Repeated addition Double Pairs How many in each group? How many altogether?

Creating groups, for example:

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Equal groups. In this case, 3 lots of 3 = 9.

Numicon

Cuisenaire

Objects

Drawing

Billingshurst Primary School Progression in Calculations Early multiplication



Practical activities, for example lining up in pairs:

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Billingshurst Primary School Progression in Calculations Early multiplication continued



Practical problem solving, for example:

45

Lego features multiplication in the number of studs on each brick .




Using resources, such as:

46

In this example, showing three lots of six, or, 3 x 6 = 18

Cuisenaire

Bead string

Numicon




Using resources, such as a number line:

47

In this example, showing three lots of six, or, 3 x 6 = 18

Billingshurst Primary School Progression in Calculations Next Steps in Multiplication – REPEATED ADDITION

Underpinning ideas Written methods

The children will already understand the ideas behind addition. They will use this knowledge to help them, identifying that multiplication is adding the same number on again and again – repeated addition. The children will use jottings and diagrams.

For example, 3 x 4 4 + 4 + 4 3 x 4 is 4 + 4 + 4 = 12 or 3 lots of 4 or 4 x 3 This can be shown on a number line: Or a bead string:

0 1 2 3 4 5 6 7 8

+4

9 10 11 12

+4 +4

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Billingshurst Primary School Progression in Calculations Multiplication – COMMUTATIVE LAW


The children should know the commutative law of multiplication. This means that the numbers in the multiplication number sentence can be written either way around and the outcome will be the same. This is also true for addition but not for subtraction or division. See those sections for more information.

For example: 3 x 4 = 12 4 x 3 = 12 Therefore, 3 x 4 = 4 x 3 This is how it would look a number line: Remember, the multiplication symbol, X, can be said out loud as ‘lots of’.

0 1 2 3 4 5 6 7 8

+4

9 10 11 12

+4 +4

+3 +3 +3 +3

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Billingshurst Primary School Progression in Calculations Multiplication – ARRAYS


Arrays allow the children to model a multiplication problems. They are useful as they form the basis of the grid method.

For example, 8 x 4 This array not only shows that 8 x 4 is 32 but also reinforces the commutative law of multiplication by showing that 4 x 8 is also 32.

4 x 8 = 32

8 x 4 = 32

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Billingshurst Primary School Progression in Calculations Multiplication – PARTITIONING


In tandem with arrays, partitioning is an important method to understand in order to enjoy success when using the grid method. Partitioning also helps the children to understand how multiplication works, particularly when multiplying TU x TU and greater. In this way, partitioning is the basis of long multiplication and allows the children a method to check what they have done.

Partitioning means considering the value of each digit in the multiplicand and the multiplier. For example: 24 x 6 24 can be partitioned in to 20 and 4 before being multiplied by 6 Therefore, 24 x 6 is the same as (20 x 6) + (4 x 6) 20 x 6 = 120 4 x 6 = 24 120 + 24 = 144 So 24 x 6 = 144 Considering 54 x 12 54 is partitioned as 50 and 4. 12 is partitioned as 10 and 2. All elements of the multiplicand must be multiplied by all parts of the multiplier. So 54 x 12 = (50 x 10) + (50 x 2) + (4 x 10) + (4 x 2) 54 x 12 = 500 + 100 + 40 + 8 = 648

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Billingshurst Primary School Progression in Calculations PARTITIONING (particularly for Grid Method) Consider 42 x 6 If this was set out as an array you would get: Essentially 42 lots of 6. However, you could also think of this as: Now we have 40 lots of 6 and 2 lots of 6. We have partitioned 42 lots in to 40 lots and 2 lots. This will help us to calculate the answer using grid method, by completing the grid and replacing the dots with numbers.

42

6

40

6

2

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Billingshurst Primary School Progression in Calculations MULTIPLYING AND DIVIDING BY 10, 100, 1000...

Underpinning ideas What this looks like

Confidently multiplying and dividing by 10, 100, 1000 and beyond brings benefits to a mathematician in both mental and written work across a range of concepts and ideas. It is vitally important that the children are taught and understand the mathematically correct vocabulary and explanations for what happens when multiplying and dividing by 10 etc. Multiplying by 10 does not mean that mathematicians ‘put a zero on the end’ or ‘add a zero’. Multiplying by 10 does mean that each digit in a number gets 10 times larger and therefore moves one place value column to the left. A place value holder zero may then be needed. Dividing by 10 does not mean that the decimal point moves. The decimal point should be treated as another place value column and these do not move. Dividing by 10 does mean that each digit in a number gets 10 times smaller and therefore moves one place value column to the right.

73 x 10 306 x 10 742 ÷ 10 50.9 ÷ 10

Place Value Columns

Th H T U t h

7 3

7 3 0

3 0 6

3 0 6 0

7 4 2

7 4 2

5 0 9

5 0 9

Multiply by Effect on all digits

Divide by Effect on all digits

x 10 1 place value column larger

÷ 10 1 place value column smaller

x 100 2 place value columns larger

÷ 100 2 place value columns smaller



A place holder zero is required in the Units column.

Billingshurst Primary School Progression in Calculations Multiplication without decimal places – GRID METHOD


Grid method allows you to see the individual calculations that make up the complete multiplication. This means you can check the correctness of your answer more easily. A secure understanding of multiplying or dividing by 10, 100, 100 etc allows the children to solve complex calculations using only their knowledge of multiplication tables up to x9. For example, 40 x 6 can be considered as : 4 x 6 then x 10 This is because 40 is 10 times bigger than 4. 4 x 6 = 24 24 x 10 = 240

Grid method – multiplying TU x U As explained earlier in the section on partitioning, 42 x 6 can also be written 40 x 6 and 2 x 6. Grid method helps children to understand this and multiply by using a frame (the grid) and following a pattern. First, children draw the grid. This is made up by creating a column for each digit in the multiplicand and a row for each digit in the multiplier. In our example, we have two columns (42) and one row (6). Next, children must partition any numbers with more than one digit. In our example 42 becomes 40 and 2. As 42 is the multiplicand, 40 and 2 are placed in the columns. As 6 is the multiplier, this goes in the row.

Now, carry out the calculations. Box A contains the answer to 40 x 6.

Box B contains the answer to 2 x 6.

Now add the answers together to show that 42 x 6 = 252

Box A Box B

240 + 12

252 54

Billingshurst Primary School Progression in Calculations Multiplication without decimal places GRID METHOD examples

HTU x U 372 x 7 TU x TU 45 x 23 HTU x TU 911 x 68

2100 490

+ 14

2604

1

800 100

+ 120 15

1035

1

54000 7200

+ 600 80 60

8

61948

1 1 55

Billingshurst Primary School Progression in Calculations Multiplication with decimal places – GRID METHOD with DECIMALS


Grid method allows you to see the individual calculations that make up the complete multiplication. This means you can check the correctness of your answer more easily. A secure understanding of multiplying or dividing by 10, 100, 100 etc allows the children to solve complex calculations using only their knowledge of multiplication tables up to x9. For example, 40 x 6 can be considered as : 4 x 6 then x 10 This is because 40 is 10 times bigger than 4. 4 x 6 = 24 24 x 10 = 240

Grid method: multiplying U x U.t U x U.t 3 x 4.7 Set out the grid by partitioning the numbers in exactly the same way as you would for a calculation with no decimal places. 4.7 can be partitioned as 4 and 0.7. Here, the calculations are 3 x 4 and 3 x 0.7 3 x 4 = 12 Now you can use your knowledge of dividing by 10 to help you, so calculate 3 x 7 instead of 3 x 0.7 3 x 7 =21 But 7 is ten times bigger than 0.7 so you must divide the answer by 10 21 ÷ 10 = 2.1

12.0

+ 2.1

14.1 56

Billingshurst Primary School Progression in Calculations Multiplication with decimal places GRID METHOD examples

When completing a grid method calculation by adding, you must ensure to set out the digits in the correct place value columns, as in the examples below: TU x U.t 49 x 5.6 TU.t x U.t 37.8 x 2.1

200.0 45.0

+ 24.0 5.4

274.4

1

60.00 14.00

+ 1.60 3.00 0.70 0.08

79.38 1

HTU .t

TU .th

57

Billingshurst Primary School Progression in Calculations Multiplication – LONG MULTIPLICATION


Long multiplication essentially mirrors the process used for grid method. However, the calculations are presented vertically rather than in a grid. Each individual calculation is shown in a row so the children can check their calculations thoroughly.

For example: 136 x 25 Remember, this calculation can be partitioned as; (100 x 20) + (100 x 5) + (30 x 20) + (30 x 5) + (6 x 20) + (6 x 5) Long multiplication shows this in the following format: 136 x 25 = 3400 As you can see, this is very similar to grid method.

136 X 25

30

150 500 120 600

+ 2000

3400

5 x 6 5 x 30 5 x 100 20 x 6 20 x 30 20 x 100

58

Billingshurst Primary School Progression in Calculations Multiplication – SHORT MULTIPLICATION


Short multiplication differs from long multiplication in the presentation of the calculations. The calculations are completed in fewer rows, one row for each digit in the multiplier. This means the children need to carry confidently and accurately. Consistency is very important as this a less easy method to double check as the individual calculations are less obvious.

136 x 25

H TU

1 3 6 X 2 5

6 8 0

2 7 2 0

3 4 0 0

3 1

1

1

Row A - This row shows the calculations for 5 x 6, 5 x 30 and 5 x 100. 5 x 6 = 30, the zero units are placed in the Units column and the three tens are carried (represented in red) in the tens column. 5 x 30 = 150, to which the three tens carried must be added. This equals 180 so an eight is placed in the Tens column and one hundred is carried. 5 x 100 = 500 to which the one hundred that was carried is added, giving a six in the Hundreds column.

Row A

Row B

Row C

Row B - This row shows the calculations for 20 x 6, 20 x 30 and 20 x 100. 20 x 6 = 120, the zero units are placed in the Units column and two tens in the Tens column and one hundred is carried. 20 x 30 = 600, to which the one hundred carried must be added. This equals 700. 20 x 100 = 2000 so a two is placed in the Thousands column. Row C – Rows A and B are added to give the answer.

1

59

Billingshurst Primary School Progression in Calculations DIVISION Contents page 61 Early division pages 62 - 64 Grouping page 65 Grouping with remainders page 66 Being careful with remainders page 67 Grouping with bigger numbers page 68 Being careful with real world division problems page 69 Setting out a division problem page 70 Dividing by 10, 100 and 1000 pages 71 & 72 Chunking method (long division) page 73 Short division page 74 Remainders presented as fractions page 75 Remainders presented as decimals

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Billingshurst Primary School Progression in Calculations Early division

The Language of Division Methods

÷ Share Share equally Share out One each, two each, three each… Groups of Equal groups of Group in pairs, threes… tens Divide Divided by Divided into Divide up Left Left over Is there enough for....each?

Practical sharing activities, which can include sharing money:

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Children sharing out pieces of fruit, making sure everyone gets the same amount.

Billingshurst Primary School Progression in Calculations Next steps in division – REPEATED SUBTRACTION

Underpinning ideas Method

As the children gain further experience in division, the emphasis is now on grouping rather than sharing. Grouping requires the children to understand repeatedly subtracting the same number to or from a starting point. Grouping using repeated subtraction as a method of division is very similar to using grouping for multiplication.

This method uses Cuisenaire and a number track to show the division visually. For example, 20 ÷ 5 Next, count back from the dividend in groups of the divisor until zero is reached. 20 has been grouped in to 5s and you can see there are 4 of these groups, so; 20 ÷ 5 = 4 The division symbol, ÷ (called an obelus), can be replaced by different words and phrases, including ‘divided in to groups of’ like this: 20 divided in to groups of 5 = 4

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Billingshurst Primary School Progression in Calculations Next steps in division – REPEATED SUBTRACTION


As the children gain further experience in division, the emphasis is now on grouping rather than sharing. Grouping requires the children to understand repeatedly subtracting the same number to or from a starting point. Grouping using repeated subtraction as a method of division is very similar to using grouping for multiplication.

This method uses a number line to show the division visually. For example, 20 ÷ 5 Children draw an empty number line and place the dividend at the right hand end. Next, count back from the dividend in groups of the divisor until zero is reached. 20 has been grouped in to 5s and you can see there are 4 of these groups, so; 20 ÷ 5 = 4 The division symbol, ÷ (called an obelus), can be replaced by different words and phrases, including ‘divided in to groups of’ like this: 20 divided in to groups of 5 = 4

20

20

15 10 5 0

-5 -5 -5 -5

63

0

Billingshurst Primary School Progression in Calculations Division – REPEATED SUBTRACTION EXAMPLES

14 ÷ 2 How many groups of 2? There are 7 groups of 2. Therefore, 14 ÷ 2 = 7 25 ÷ 5 How many groups of 5? There are 5 groups of 5. Therefore, 25 ÷ 5 = 5 60 ÷ 10 How many groups of 10? There are 6 groups of 10. Therefore, 60 ÷ 10 = 6

60 50 40 30 20 10 0

-10 -10 -10 -10 -10 -10

25 20 15 5 10

12 10 8 6 4 2

-2

0

0 14

-2 -2 -2 -2 -2 -2

-5 -5 -5 -5 -5

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Billingshurst Primary School Progression in Calculations Division – REPEATED SUBTRACTION WITH REMAINDERS


The number line method can also be used to solve problems where there is a remainder.

Repeated subtraction with a remainder. This method uses a number line to show the division visually. For example, 23 ÷ 10 children draw an empty number line and again place the dividend at the right hand end Next, count back from the dividend in groups of the divisor. However, this time it is not possible to reach zero, as taking another group of 10 away would reach a negative number. The remainder in this case is 3, which is the difference between the final number reached and the target number, in this case zero. So, 23 ÷ 10 = 2 remainder 3 In shorthand this is written as 23 ÷ 10 = 2 r 3

23

23

13 3 0

-10 -10

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Billingshurst Primary School Progression in Calculations Division – *** BEING CAREFUL WITH REMAINDERS ***


The remainder should never be bigger than the divisor.

Grouping – repeated subtraction with a remainder. The remainder should never be bigger than the divisor. For example: 32 ÷ 5 32 ÷ 5 cannot equal 5 r 7 because the remainder, 7, is bigger than the divisor, 5! This means we can take away another group of 5. How many groups of 5 are there? There are 6 groups of 5. There is a remainder of 2. So, 32 ÷ 5 = 6 r 2

32 27 22 17 12 7

-5 -5 -5 -5 -5

0

32 27 22 17 12 7

-5 -5 -5 -5 -5 -5

2 0

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Billingshurst Primary School Progression in Calculations Division – REPEATED SUBTRACTION WITH BIGGER NUMBERS


Grouping is a good method because every step of the working out process can be seen. This method can also be used with bigger numbers. There methods can speed up the calculation process considerably but rely on an excellent working knowledge of multiplication tables. This is just another example of where multiplication tables help a mathematician and why they should be learned off by heart.

For example, 72 ÷ 5 There are 14 groups of 5 with a remainder of 2. Therefore 72 ÷ 5 = 14 r 2 However, with practise and considering their multiplication table knowledge, the children can speed up this process. Here you can see that to reach the answer, a total of 14 lots of 5 have been taken away. Firstly, a chunk of 10 lots of 5 was taken away. Then 4 individual lots of 5 were taken away to leave the remainder of 2. This could be speeded up even further: 72 ÷ 5 = 14 r 2

72 22

-50 (take away 10 lots of 5)

0

-5 -5 -5 -5

2

72 22


0 2


67

72

-5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5

67 62 57 52 47 42 37 32 27 22 17 12 7 2 0

Billingshurst Primary School Progression in Calculations Division *** BEING CAREFUL WITH ROUNDING UP AND ROUNDING DOWN ***


Division is commonly used to solve real world problems. Therefore, when faced with a remainder the children must decide whether it makes more sense to round up or round down to the nearest integer.

For example: One packet of biscuits contains 10 biscuits and I have 23 biscuits. How many packets can I make? children can calculate using the following method: Each jump of 10 represents one packet of biscuits, therefore two packets can be made. But what to do with the remainder of 3 biscuits? Here, the children will need to use common sense. If packets of biscuits are supposed to contain 10 biscuits, would they be happy to buy a pack and find out it only contained 3 biscuits? So in this case we would round down and say that only 2 packets can be made. Therefore the answer is 2 packets of biscuits. Conversely, the children may solve a problem where rounding up makes sense. Imagine a restaurant that has tables which seat 4 people. 38 people arrive for dinner. How many tables are needed? This time we will use repeated addition. Nine jumps of 4 in this case means nine tables of 4 people. Nine tables would seat 36 people. However, that remainder of 2 is two people who haven’t got anywhere to sit for dinner! They are not going to be very happy about that, so we need an extra table. In this case, it doesn’t matter that the tenth table only has two people sitting at it, we just have to make sure the are sitting down. So the answer is 10 tables.

23 13 3 0

-10 -10

8 4 0 12 16 20 28 24 32 36 38

+4 +4 +4 +4 +4 +4 +4 +4 +4

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Billingshurst Primary School Progression in Calculations Division *** BEING CAREFUL WITH SETTING OUT THE PROBLEM***

Underpinning ideas Written examples

The commutative law of addition. The commutative law of multiplication. They sound a bit complicated! Actually, all ‘commutative law’ means is that you can write the problem down with the numbers in any order and the answer will be the same. For example: 3 + 1 gives the same answer as 1 + 3 4 x 5 gives the same answer as 5 x 4

However, commutative law DOES NOT APPLY TO DIVISION PROBLEMS! 20 ÷ 5 does not give the same answer as 5 ÷ 20. If you consider using ‘shared between’ in place of the division symbol, we could write these as real world problems: 20 sweets shared between 5 people (20 ÷ 5) There are 4 jumps of 5 so everyone gets 4 sweets. compared with 5 sweets shared between 20 people (5 ÷ 20). In this case there are fewer sweets than people so everyone would get less than one sweet. As the children develop in mathematics, they will need to solve problems which leave an answer smaller than one, such as 5 ÷ 20, but the children must consider the context of the question when setting it out in writing.

20 15 10 5

-5 -5 -5 -5

0

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Billingshurst Primary School Progression in Calculations MULTIPLYING AND DIVIDING BY 10, 100, 1000...

Underpinning ideas What this looks like

Confidently multiplying and dividing by 10, 100, 1000 and beyond brings benefits to a mathematician in both mental and written work across a range of concepts and ideas. It is vitally important that the children are taught and understand the mathematically correct vocabulary and explanations for what happens when multiplying and dividing by 10 etc. Multiplying by 10 does not mean that mathematicians ‘put a zero on the end’ or ‘add a zero’. Multiplying by 10 does mean that each digit in a number gets 10 times larger and therefore moves one place value column to the left. A place value holder zero may then be needed. Dividing by 10 does not mean that the decimal point moves. The decimal point should be treated as another place value column and these do not move. Dividing by 10 does mean that each digit in a number gets 10 times smaller and therefore moves one place value column to the right.

73 x 10 306 x 10 742 ÷ 10 50.9 ÷ 10

Place Value Columns

Th H T U t h

7 3

7 3 0

3 0 6

3 0 6 0

7 4 2

7 4 2

5 0 9

5 0 9

Multiply by Effect on all digits

Divide by Effect on all digits

x 10 1 place value column larger

÷ 10 1 place value column smaller





A place holder zero is required in the Units column.

Billingshurst Primary School Progression in Calculations Division – CHUNKING METHOD (Long Division)


The chunking method of division essentially mirrors the grouping method using repeated addition (multiplication) on a number line. In this case, the horizontal number line formation is replaced by a vertical arrangement.

67 ÷ 5 Compare this with an earlier method:

71

6 7 5

50 10 x 5 -

17 15 3 x 5 -

2

The dividend (67) is placed inside the ‘bus stop’ shape, with the divisor (5) outside to the left

Again, children use their multiplication knowledge to help them. What is a jump, or ‘chunk’ in the 5 times table that will get close to 67? 10 x 5 should get us close. So we have already got to 50, but how many left to get to 67? We subtract to find out. This process continues until either the dividend (67) is reached or there is a remainder. Remember, the remainder should not be bigger than the divisor.

17 67

10 x 5 (10 lots of 5)

3 x 5 (3 lots of 5)

2

Billingshurst Primary School Progression in Calculations Division – CHUNKING METHOD EXAMPLES

TU ÷ U HTU ÷ U HTU ÷ TU 98 ÷ 7 98 ÷ 7 = 14

442 ÷ 13 It is helpful in this type of question to write down an unknown multiplication table up to 10 lots to aid in solving the problem. For example: 13 26 39 52 65 78 91 104 117 130 442 ÷ 13 = 34

546 ÷ 4 546 ÷ 4 = 136 r 2

98 7

70 10 x 7 -

28 28 4 x 7 -

0

546 4

400 10 0 x 4 -

146 100 25 x 4 - 442 13

130 10 x 13 -

312 260 20 x 13 -

46 40 10 x 4 -

6 4 1 x 4 -

2 52 4 x 13 - 0

52

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Billingshurst Primary School Progression in Calculations Division – SHORT DIVISION


Short division is a relevant method for children who have a solid grasp of the fundamental ideas behind division and the stages which lead to this method. In particular, children will continue to require excellent multiplication knowledge and be confident in their understanding of place value. The reason for this is that short division does not follow many of the patterns we see in written calculation methods for the other operators. For example, short division begins by considering the highest place value column, not the lowest, and does not assign the true value to each column but rather uses the digit as a value.

132 ÷ 7 You can see the difference to most other methods. The first calculation is 1 divided by 7 which does not leave an integer and so the 1 is carried to the next digit. The 1 and 3 are considered to represent 13. Then 7 ‘goes in to’ 13 once with a remainder of 6. The 6 is carried so we now consider how many times 7 ‘goes in to’ 62. This is 8 times. However, 7 lots of 8 are 56 so we have a remainder of 6. As a good mathematician knows, the digit 1 really represents 100 in this number but this method does not make that clear. Any mathematician who uses this method should be able to give a detailed explanation of how and why it works. If they cannot it suggests they have learned the method by rote which will lead to problems further down the line when they need to understand the mathematics behind the method to progress, or within other contexts.

73

1 3 2 7

0 1 8 r 6 6 1

Billingshurst Primary School Progression in Calculations Division – REMAINDERS REPRESENTED AS FRACTIONS


In most circumstances, it is acceptable to show a remainder as just that. However, on occasion it might be more useful for the mathematician to show a remainder as a fraction.

Let us consider again 67 ÷ 5 67 ÷ 5 = 13 r 2 The fraction can be created by using the remainder as the numerator and the divisor as the denominator. In this case the remainder is 2 and the divisor is 6, so the fraction becomes: Therefore 67 ÷ 5 = 13 2/5

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17 67

10 x 5 (10 lots of 5) 3 x 5

(3 lots of 5)

2

5 Divisor becomes the

denominator

Remainder becomes the

numerator

2

Billingshurst Primary School Progression in Calculations Division – REMAINDERS REPRESENTED AS DECIMALS


In most circumstances, it is acceptable to show a remainder as just that. However, on occasion it might be more useful for the mathematician to show a remainder as a decimal.

There are two main ways to do this. Firstly the children could turn a remainder in to a fraction and then convert the fraction in to a decimal. For example, 67 ÷ 5 = 13 r 2 = 13 2/5 = 13.4 Alternatively, it is possible to use either long division (chunking method) or short division to create an answer with digits in the decimal places. 132 ÷ 7 Rather than stop at the units place, we can use a decimal point and continue in to the decimal places: You can see that to the right of the decimal point in the tenths place value column there is a place holder zero so that the remainder 6 can be carried. 7 in to 60 goes eight times with a remainder of 4 which is carried to another place holder zero in the hundredths place. 7 in to 40 goes 5 times with a remainder of 5 and this process can continue as far as accuracy dictates. The short division method remains the same but simply occurs in the decimal place value columns.

75

1 3 2 0 0 7

0 1 8 8 5 6 6 4

1 3 2 7

0 1 8 r 6 6 1

1

Billingshurst Primary School Progression in Calculations Notes

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