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Nonsuch Primary School
Calculation Policy
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Kate Mole Nonsuch Primary School
2014
Page 1
Introduction
One of the aims of the Primary Mathematics Curriculum 2014 is:
To become fluent in the fundamentals of mathematics, including
through varied and frequent practice with increasingly complex
problems over time, so that pupils develop conceptual
understanding and the ability to recall and apply knowledge rapidly
and accurately.
In short, this means that children need to know not just what to do, but
why they are doing it.
At Nonsuch, our aim is that all pupils will develop arithmetic proficiency
– they will be able to do maths easily and at speed. Arithmetic
proficiency does not come from simply committing number facts and
procedures to memory, but from a growing understanding of numbers
and the relationships between them. This policy, and the teaching of
calculation throughout the school, focuses on developing procedural
fluency AND the conceptual understanding in parallel.
 Procedural fluency is knowledge of procedures, knowledge of
when and how to use them appropriately, and skill in performing
them flexibly, accurately and efficiently.
 Conceptual understanding is comprehension of mathematical
concepts which are developed through establishing connections
between procedures, manipulatives (hands on practical
resources), visual representations (images and pictures) and prior
knowledge.
Teaching of calculation
Mental strategies:
Written methods of calculations are based on mental strategies. Each of
the four operations builds on mental skills which children must have a
secure grasp of in the Early Years and Key Stage 1.
Jottings and informal methods:
Mental strategies provide the foundation for jottings and informal written
methods of recording which are taught in Key Stage 1. Skills need to be
taught in a way that will give the pupils conceptual understanding in
Kate Mole Nonsuch Primary School
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Page 2
addition to procedural knowledge. They need to be practised and
reviewed regularly.
Formal written methods:
The above skills lead on to formal written methods of calculation which
are taught from Year 3 upwards. Formal written methods (or laying
calculations out in a columnar format) must not occur in Key Stage 1.
Again, the teaching of formal written methods should be taught in a way
that will give the pupil secure conceptual understanding in addition to
procedural knowledge.
The aim of this policy is to inform teaching staff and parents of:
 The progression in the teaching of calculations from Reception to
Year 6
 The methods that will be used to teach calculation of the four
operations
 The importance of developing procedural fluency AND conceptual
understanding in parallel
How to develop procedural fluency and conceptual understanding in
parallel
Strategies and methods for mental and written calculation need to be
supported by familiar manipulatives (hands on practical equipment) and
visual representations, in order to develop conceptual understanding at
the same time as procedural fluency.
Throughout this document, teaching strategies and methods will include
the use of a variety of manipulatives and visual representations. These
(or any other relevant manipulatives and / or visual representation)
MUST be used when teaching and modelling methods of mental and
written calculation, AND they must be accessible to ALL children during
independent or guided learning.
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The following strategies will be used to teach all four operations and will
be referred to throughout this policy:
Number Rods
Number rods are used to represent numbers. They:
 Have a strong visual and tactile appeal that relates well to how
pupils learn;
 Can be thought of as coloured bits of a number line that you can
move around;
 Are powerful images that support pupils to make relationships
between and within number more noticeable.
Pupils can use the number rods to
demonstrate concretely what they are
trying to say about numbers. For
example, when calculating 47 ÷ 5, a pupil
explores how many 5s make 47 (a
grouping model). By placing yellow rods
(which represent 5) along a number line,
they can demonstrate that nine 5s make
45 and there is a remainder of 2.
Bar Models
A bar model is a pictorial representation of a calculation problem that
helps a pupil to visualise the problem and to identify where the unknown
quantity is. It enables the pupils to consolidate or manipulate the
information given in the problem and to understanding the mathematical
concepts and relationships contained in it.
The pupils draw a bar model to visualise the calculation and then use
number rods or Base 10 apparatus alongside written methods to
calculate the answer.
Three steps are taken to ensure that the bar model is understood by the
pupils. Please note that the unknown is always drawn in a dotted line.
Step 1
The bar model is a pictorial representation of the number rods. It is
important that the pupils draw each bar individually so that the
mathematical relationships in the problem are clear.
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10 – 6 = ?
Step 2
The bar model is a pictorial representation of the calculation.
876 – 675 = ?
Step 3
The bar model is a pictorial representation of a problem.
A book has 876 pages. If you read 675 pages, how many pages do you
still have to read?
By drawing a model, pupils can see a problem clearly and solve it
effectively.
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Regrouping
The term ‘regrouping’ will be used to ensure consistency of language
throughout the school in all four strands of calculation.
Regrouping emphasises that the value of a number has not changed.
For example, when a pupil regroups 12 ones to 1 ten and 2 ones the
value of the number has not changed; the number has just been
regrouped in a different way.
In the process of solving a calculation, base 10 apparatus or place value
counters are manipulated in order to move information around, enabling
the pupils to understand the relationship between hundreds, tens and
ones when re-grouping to solve a calculation.
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Use of place value mats, partitioning mats and calculation mats to
support calculation
Mats of different sorts give pupils a framework and structure within which
to work. They can set out the steps for pupils to take them through the
process of going from a symbolic representation of a calculation (a
number or number sentence), to manipulating hands on resources to aid
understanding, to being able to achieve the objective.
Partitioning mat
Number rods (or base 10 equipment) are used
as a manipulative to give the children
conceptual understanding and secure
knowledge of place value in addition to
procedural knowledge.
Progression:
The expectation is that the majority of pupils will move through the
progressive stages in calculation at broadly the same pace. However,
decisions about when to progress should always be based on the
security of the pupils’ understanding and their readiness to progress to
the next stage. The progression in this document is outlined in year
groups, based on the expectations and progression in the Primary
National Curriculum for Mathematics 2014.
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Counting
Counting should take place in every year group for 3 minutes per day.
This does not have to be during the maths lesson. Ideas for counting are
provided to all teachers. The use of counting sticks, hundred squares
and other manipulatives and visual representations are encouraged.
The statutory requirements are outlined below but other forms of
counting (e.g. around the clock, using money, with decimals and
percentages) should also be delivered during daily counting sessions.
Year 1 Number and place value (statutory requirements)
-count to and across 100, forwards and backwards, beginning with 0 or 1,
or from any given number
-count, read and write numbers to 100 in numerals, count in different
multiples including ones, twos, fives and tens
Year 2 Number and place value (statutory requirements)
-count in steps of 2, 3, and 5 from 0, and count in tens from any number,
forward or backward
Year 3 Number, place value and rounding (statutory requirements)
-count from 0 in multiples of 4, 8, 50 and 100; finding 10 or 100 more or less
than a given number
-count up and down in tenths; recognise that tenths arise from dividing an
object into 10 equal parts and in dividing one-digit numbers or quantities by
10
Year 4 Number, place value and rounding (statutory requirements)
-count in multiples of 6, 7, 9, 25 and 1000
-count backwards through zero to include negative numbers
-count up and down in hundredths; recognise that hundredths arise when
dividing an object by a hundred and dividing tenths by ten
Year 5 Number, place value, approximation and estimation
(statutory requirements)
-count forwards or backwards in steps of powers of 10 for any given number up to
1 000
Kate Mole Nonsuch Primary School
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Addition
Manipulatives and Images
Counting apparatus
Number rods
Base 10 equipment
Place Value counters
Place value cards
Number tracks
Numbered number lines
Marked but unnumbered number lines
Empty number lines
Hundred square
Counting stick
Bead string
Models and Images charts
ITPs – Number Facts, Ordering Numbers, Number Grid, Counting on
and
back in ones and tens
The bar model for addition
Can you draw me a bar model to represent the question/ calculation?
e.g.
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Reception
Recognise numerals
0 to 10 and understand
the meaning of each
number by recognising
and knowing the number
rods
Count reliably up to 10
everyday objects
Find one more than a
number
One more than
three is four
What is the
number
after…?
The number after
three is four
Recognise numerals up to 20 and understand the meaning of each
number by recognising and knowing the number rods
Count in ones as they become more familiar with
using numbers on a numberline and 100 square
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Begin to relate addition
and combing two
groups of objects
Begin to use the + and
= signs to record
mental calculations in a
number sentence
Children begin to solve
problems using
doubling
Children show mental
recall of number bonds
to 10 and use these for
problem solving. Some
children show mental
recall of number bonds
to 20
Year 1
Continue with the use of number rods in order for children to consolidate their
understanding of the value of numbers and for them to develop the skills of
partitioning numbers in different ways
Can you order the rods
from smallest to
largest?
How many white rods do you
think will fit along a yellow
rod?
How many white rods will fit
along a dark green rod?
Can you find the purple and
green rod? Put them together
and find the rod that is the
same size. We can partition
the black into purple and
green (then attribute
numbers)
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Know by heart all pairs
of numbers with a total
of 10 and 20
Know that addition can
be done in any order
When adding mentally
put the biggest number
first and count on
Progression in difficulty when adding on a number line
– circle number that you are starting from and cross
out in the calculation
1 digit + 1 digit
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2 digits + 1 digit
2 digits + 2 digits
Note: for this method to be successful and embedded, children must be
able to add 10 and 1 from any given number using their knowledge of
place value and having had plenty of experience with jumping in ones
and tens on a 100 square.
Begin to partition
numbers in order to add
two 2 digit numbers
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Year 2
Continue with the use of number rods in order for children to consolidate their
understanding of the value of numbers and for them to develop the skills of
partitioning numbers in different ways.
Can you order the rods
from smallest to
largest?
How many white rods do you
think will fit along a yellow
rod?
How many white rods will fit
along a dark green rod?
Can you find the purple and
green rod? Put them together
and find the rod that is the
same size. We can partition
the black into purple and
green (then attribute
numbers)
Continue to practise the
mental recall of all pairs of
number bonds with a total of
10 and 20
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Know which digit changes
when adding 1s or 10s to any
number
Continue with partitioning two
digit numbers into their tens
and ones and extend to
partition three digit numbers
into their H T and O
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In Year 2 pupils learn to add using a blank numberline. They will record this in
their maths book and will draw the blank numberline themselves. They should
not use a ruler to draw the line.
Progression in difficulty when adding on a blank numberline.
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5. Bridging through 10 – add tens first then ones
47 + 28
+20
+5
+3
47
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2014
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Year 3, 4, 5 and 6
Formal written methods - Column method for addition
Note that the number line method should still be modelled as part of a
mental maths strategy. Pupils should progress to be able to add
mentally as they are then able to visualise a numberline in their head.
 Begin with teaching this method without ‘carrying’
 ‘Regrouping’ should be used as per the introduction to this policy,
before numbers are ‘carried’ into the next column
 Carried digits are recorded below the line using the words ‘carry
ten’ or ‘carry one hundred’, not ‘carry one’
 At each stage children ‘do then record’ before moving onto the
next stage / column. Pupils should never complete the whole
calculation practically without recording, or complete the whole
calculation in a recorded format without ‘doing’
 Children need to be careful how they set out the numbers when
calculating, especially with decimals
In Year 3 children should be taught to add numbers with up to 3 digits
using a Calculation Mat and Base 10 equipment. Children should use
equipment during independent and guided learning.
In Year 4 children should be taught to add numbers with up to 4 digits
using a Calculation Mat and Base 10 equipment, moving onto place
value counters.
In Year 5 children should be taught to add whole numbers with more
than 4 digits, modelled by the teacher using Base 10 equipment / Place
Value Counters. Children may choose or choose not to use equipment
during independent learning. Children who are not working at the Year 5
level should use equipment to develop their conceptual understanding.
In Year 6 children should develop and secure their understanding when
adding numbers with more than 4 digits and those including decimals.
Kate Mole Nonsuch Primary School
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Progression in the formal written method for addition
2 digits + 2 digits / 3 digits + 3 digits with no carrying into the tens column.
32
+ 25
Bar model:
1. Do: Lay out on the
calculation mat with Base
10 and record the
calculation at the side
2. Do: add the ones by
combining the 2 and 5
ones. Record.
3. Do: add the tens by
combining the 3 tens and 2
tens. Record.
2 digit + 2 digit / 3 digit + 3 digit with carrying into the tens column.
47
+ 5
Bar model:
Kate Mole Nonsuch Primary School
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1. Do: Lay out on the
calculation mat with Base
10 and record the
calculation at the side
2. Do: Add the units by
combining the 7 and 5 ones
to make 12 ones
3. Do: Regroup 10 ones as 1
ten and then carry into the
10s column. Record.
2 digits + 2 digits / 3 digits + 3 digits with carrying into the tens and
hundreds column.
77
+ 45
Bar Model:
Kate Mole Nonsuch Primary School
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1. Do: Layout out on the
calculation mat with Base 10
and record the calculation at
the side
2. Do: 5 ones plus 7 ones equal
12 ones so they need to be
regrouped into 1 ten and 2
ones.’ Carry’ the ten into the
tens column. Record.
3. Do: 7 tens plus 4
tens, plus the 1 ten
we carried, equal
12 tens so they
need to be
regrouped into 1
hundred and 2
tens. ‘Carry’ the
hundred into the
hundred column.
Record.
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The above teaching strategies are used to teach the following
calculations. Pupils should draw a bar model to represent the calculation
/ problem at all stages. From Year 4 upwards, once children are secure
with the use of Base 10 equipment to ensure conceptual understanding,
place value counters can be used.
676 + 558
676
+ 558
1234
1 1
Column addition with decimals
In the following examples pupils need to understand that the decimal
points are always written underneath each other when using column
addition. Pupils should have a secure understanding of place value in
order to be successful.
Adding amounts of money provides a useful context within which to
teach the addition of decimals. A calculation mat for money supports
conceptual understanding.
Kate Mole Nonsuch Primary School
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12.5
+ 23.7
36.2
12.5 + 23.7
1
123.5 + 24.6
123.5
+ 24.6
148.1
1
34.5 + 27.43
34.50
+ 27.43
61.93
1
Kate Mole Nonsuch Primary School
2014
Use zero as a
place holder
Page 23
Subtraction
Mental skills
Recognise the size and position of numbers
Count on or back in ones and tens
Know number facts for all numbers to 20
Subtract multiples of 10 from any number
Partition and recombine numbers (only the number to be
subtracted)
Bridge through 10
Manipulatives and Images
Counting apparatus
Number rods
Place value apparatus
Base 10 equipment
Place value counters
Number tracks
Numbered number lines
Marked but unnumbered numberlines
Hundred squares
Empty numberlines (not to be drawn with a
ruler)
Counting stick
Bead strings
ITPs – Number facts, counting on and back in
ones and tens, Difference
Bar Model for subtraction
Kate Mole Nonsuch Primary School
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Reception
Begin to count backwards
in familiar contexts such
as number rhymes or
stories
Continue to count back in
ones from any given
number
Begin to relate subtraction
to ‘taking away’
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Find one less than any
given number
What is the number
before 6?
Count back in ones
Year 1
Begin to use the – and =
signs to record mental
calculations in a number
sentence
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Know by heart subtraction
for numbers up to 20
Subtract single digit numbers by
counting back along a marked
number line.
Kate Mole Nonsuch Primary School
2014
15 – 8 = 7
Page 27
Begin to find the difference by
counting on from the smallest
number
Use number rods to calculate the
difference:
What’s the difference between 8 and
10? The difference is 2.
Count along a number line from the
smallest number to the largest
number (summer term). Also show
with number rods as a practical
support.
15 – 7 = 8
Year 2 – subtracting using a numberline
Partition one and two digit numbers
using partitioning mats and apparatus.
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Subtract one from any
given number and know
which digit changes .
Subtract ten from any
given number and know
which digit changes.
18 – 1 = 17
28 – 10 = 18
Use a blank numberline to
subtract numbers that do not
bridge through 10. Partition the
number to be subtracted.
Children jump back in tens and
ones. For this to be successful,
children must be confident with
the above.
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Page 29
Use a blank numberline to
subtract numbers that
bridge through 10.
Partition the number to be
subtracted. Children jump
back in tens and ones
(they may combine their
jumps).
74 – 27 = 47
Use a blank numberline to
find the difference – how
many numbers are in
between?
29
Year 3, 4, 5 and 6
Formal written methods - Column method for subtraction
Note that the number line method should still be modelled as part of a
mental maths strategy. Pupils should progress to be able to subtract
mentally as they are then able to visualise a numberline in their head.
When calculating with time and finding time differences, the numberline
method should be used EVERY TIME.
 Begin with teaching this method without ‘carrying’
 ‘Regrouping’ should be used as per the introduction to this policy
 At each stage children ‘do then record’ before moving onto the
next stage / column. Pupils should never complete the whole
Kate Mole Nonsuch Primary School
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calculation practically without recording, or complete the whole
calculation in a recorded format without ‘doing’
 Children need to be careful how they set out the numbers when
calculating, especially with decimals
 Children should be encouraged to use inverse operations to check
if their answer is correct. This gives them the opportunity to
practise both operations (addition and subtraction) at the same
time. Explicit teaching needs to point out that if they add the
bottom number to the answer they should end up with the number
they started with. Always support with Base 10 equipment to
secure conceptual understanding.
In Year 3 children should be taught to subtract numbers with up to 3
digits using a Calculation Mat and Base 10 equipment. Children should
use equipment during independent and guided learning.
In Year 4 children should be taught to subtract numbers with up to 4
digits using a Calculation Mat and Base 10 equipment, moving onto
place value counters.
In Year 5 teaching should be taught to subtract whole numbers with
more than 4 digits, modelled by the teacher using Base 10 equipment /
Place Value Counters. Children may choose whether or not to use
equipment during independent learning. Children who are not working at
the Year 5 level should use equipment to develop their conceptual
understanding.
In Year 6 children should develop and secure their understanding when
subtracting numbers with more than 4 digits and those including
decimals.
Kate Mole Nonsuch Primary School
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2 digits - 2 digits / 3 digits - 3 digits with no regrouping. Model writing in
both the horizontal and vertical forms.
37 – 16
37
-16
Bar model:
1. Do: Lay out on the
calculation mat with
Base 10 and record the
calculation at the side.
2. Do: Subtract 6 ones and
remove them. Record.
3. Do: subtract 1 ten and
remove it. Record. How
many are left?
Kate Mole Nonsuch Primary School
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2 digits - 2 digits / 3 digits - 3 digits with regrouping tens into 10 ones.
432 - 124
bar model:
1. Do: Lay out 432 on the
calculation mat with Base
10 and record the
calculation at the side
2. Do: discuss how 4 ones
cannot be subtracted from 2
ones. Regroup a ten as 10
ones and place in the ones
column. Record.
3. Do: subtract the units –
remove them. Record.
4. Do: subtract the tens –
remove them. Record.
5. Do: subtract the hundreds –
remove them. Record.
6. Check using the inverse.
Use Base 10 to support
conceptual understanding
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3 digits - 3 digits with regrouping the tens into ones and the hundreds
into ten 10s.
532-254 =
532
- 254
Bar Model:
1. Do: Layout out on the
calculation mat with
Base 10 and record the
calculation at the side
2. Do: discuss subtracting
the ones. Regroup a ten
as 10 ones. Record.
3. Do: Subtract the ones –
remove them. Record.
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4. Do: Discuss subtracting
the tens. Regroup 1
hundred as 10 tens to
make 12 tens altogether.
Record.
5. Do: subtract the tens –
remove them. Record.
6. Do: discuss and subtract
the hundreds – remove
them. Record.
7. Prove the answer is
correct by using the
inverse. Use Base 10 to
support conceptual
understanding.
Pupils should draw a bar model to represent the calculation / problem at
all stages. Once children are secure with the use of Base 10 equipment
to ensure conceptual understanding, place value counters can be used
from Year 4 upwards.
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Column subtraction with decimals
In the following examples pupils need to understand that the decimal
points are always written underneath each other when using column
subtraction. Pupils should have a secure understanding of place value in
order to be successful.
Subtracting amounts of money provides a useful context within which to
teach the subtraction of decimals. A calculation mat for money supports
conceptual understanding.
£2.43 - £1.29 =
1. Do: discuss subtracting
the ones. Regroup a ten
as 10 ones. Record.
3 1
£2.43
- 1.29
_____
2. Do: Subtract the ones –
remove them. Record.
3 1
£2.43
- 1.29
4
Kate Mole Nonsuch Primary School
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3. Do: Subtract the tens –
remove them. Record.
4. Subtract the hundreds –
remove them. Record.
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Multiplication
Mental Skills
Recognise the size and position of
numbers
Count on in different steps 2s, 5s, 10s
Double numbers up to 10
Recognise multiplication as repeated addition
Quick recall of multiplication facts
Use known facts to derive associated facts
Multiplying by 10, 100, 1000 and understanding the effect
Multiplying by multiples of 10
Models and Images
Counting apparatus
Base10 equipment
Place Value counters
Place value apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines
Multiplication squares
Counting stick
Bead strings
Models and Images charts
ITPs – Multiplication grid, Number Dials, Multiplication Facts
Bar model for multiplication:
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Year 1
Count in steps of 2s 5s and 10s
forward and back from 0 and from any
of its multiples using the 100 square.
Take the opportunity to discuss
patterns that are recognized.
Other resources that aid counting
1. Using laminated sheets with circles (groups) on them, children group objects using
the correct mathematical vocabulary.
2. Children begin using jottings of simple multiplication with the associated vocabulary.
They begin by drawing the
number of groups, then draw
dots inside the circles. They
count the number of dots they
have altogether to get the
answer.
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3. Children are exposed to the different ways in which multiplication can be
expressed using concrete materials and linking it to real life situations. They begin
to understand that repeated addition can also be expressed as multiplication using
concrete materials.
Expressing multiplication as
repeated addition. Use number
rods to support understanding.
Expressing multiplication as
arrays.
4. Children begin to commit multiples of 2, 5, 10 to memory and use these facts to solve
problems.
There are 10 spiders. How many
legs do they have altogether?
8 x 10 = 80
When Peter behaves well in
school he gets 2 sweets at the
end of the day. If he behaves
well for 5 days, how many
sweets will he get altogether?
There are 4 flower beds in a
garden. Each flower bed has 3
flowers. How many flowers are in
the garden altogether?
Kate Mole Nonsuch Primary School
2014
Page 40
Year 2
Counting Choir
Count in steps of 2s, 3s, 5s, 10s
and 20s forward and back from 0
and from any of its multiples
using the 100 / 200 square. Take
the opportunity to discuss
patterns that are recognised.
Know doubles and their
corresponding halves. Extend to
partitioning numbers, then double and
halve.
1. Children continue to use jottings of simple multiplication with the associated
vocabulary. Those who still find it difficult use laminated mats with circles to group
concrete objects or arrays.
They begin by drawing the
number of groups, then draw the
number of dots inside the circles.
They count the number of dots
they have altogether to get the
answer.
Kate Mole Nonsuch Primary School
2014
Page 41
2. Children use number rods as a practical way to develop understanding of
multiplication as repeated addition. This also supports them in working horizontally
along a number line which will then progress to jumping along numberlines in equal
groups.
4 x 5 = 20
3. Teach using apparatus and jottings when multiplying multiples of 10 by putting /
drawing Base 10 in each of the groups
They begin by drawing the
number of groups, then put /
draw the 10s inside the circles.
They count the number of 10s
using their knowledge of
counting in 10s to obtain the
answer.
4. Teach jumping on a marked numberline in multiples of 2, 3, 5 and 10
This method requires children
to keep the jumps equal in size
as they count the number of
jumps. This is a challenging
process however it further
embeds the understanding of
repeated addition. The
constant re-enforcement of
vocabulary ‘groups of’ is very
important.
5. As children become confident with counting in multiples of 2, 3, 5, 10, they begin to
use the empty numberline to solve multiplication problems.
In this method there are strong
links with the activity of
counting choir using the 100 /
200 squares and the
recognition of patterns with
each of the multiples. Children
write their own number after
each jump they make.
Kate Mole Nonsuch Primary School
2014
Page 42
6. They further develop their skills of problem solving using multiplication and begin to
relate it to the area of a rectangle / square
Children investigate the
number of multilink cubes
needed to create a block with a
given number of length and
width. This further reinforces
the commutative law of
multiplication.
Kate Mole Nonsuch Primary School
2014
Page 43
Year 3, 4, 5 and 6
Grid multiplication alongside short and long
multiplication
2 digit x 1 digit (no regrouping) using apparatus.
13 x 3
Bar model
1. Do: Model how 13 can
be partitioned into 10
and 3 using number
rods and that we need 3
lots of these. Record
this on a grid with the
vertical layout
.alongside. .
13
x 3
x
3
2. Do: Multiply the ones.
3 x 3 – refer to number
rods and times tables
facts. Record on the grid
and the vertical format.
x
3
3. Do: Multiply the tens.
3 x 10 – refer to number
rods and times tables
facts. Record on the grid
and the vertical format.
.
4. Do: add up the tens and
ones, referring to the
apparatus and the grid.
30 + 9 = 39. Record the
answer on the vertical
format.
.
x
3
Kate Mole Nonsuch Primary School
x
3
2014
10
10
10
30
10
30
3
3
9
3
9
3
9
13
x 3
9
13
x 3
9
30
13
x 3
9
30
39
Page 44
2 digit x 1 digit (with regrouping) using apparatus. Note the number rods or place value
counters can be used (see NCETM video)
24 x 6
Bar model
1. Do: Model how 24 can be partitioned into
20 and 4 using number rods and that we
need 6 lots of these. Record this on a grid
with the vertical layout alongside.
x
6
2. Do: Multiply the ones.
6 x 4 = 24 – refer to
number rods and times
tables facts. Record on
the grid and the vertical
format.
x
6
3. Do: Multiply the tens.
6 x 20 = 120 – refer to
number rods and times
tables facts. Record on
the grid and the vertical
format.
.
x
6
4. Do: add up the tens and
ones, referring to the
apparatus and the grid.
Record the answer on the
vertical format.
.
Kate Mole Nonsuch Primary School
x
6
2014
20
20
20
120
20
120
4
4
24
24
x 6
24
x 6
24
4
24
24
x 6
24
120
4
24
24
x 6
24
120
144
Page 45
3 digits x 1 digit using place value counters.
164 x 3
Bar model
1. Do: Model how 164 can be partitioned
into 100 and 60 and 4 using number
place value counters and that we need
3 lots of these. Record this on a grid
with the vertical layout alongside.
x
3
2. Do: Multiply
the ones.
3 x 4 = 12 –
refer to place
value counter
and times
tables facts.
Record on the
grid and the
vertical format.
3. Do: Multiply
the tens.
3 x 60 = 180 –
refer to place
value counters
and times
tables facts.
Record on the
grid and the
vertical format.
100
x
3
x
3
Kate Mole Nonsuch Primary School
100
100
2014
60
4
60
60
180
164
x 3
4
12
4
12
164
x 3
12
164
x 3
12
180
Page 46
4. Do: Multiply
the hundreds.
3 x 100 = 300
– refer to place
value counters
and times
tables facts.
Record on the
grid and the
vertical format.
x
3
100
300
60
180
4
12
164
x 3
12
180
300
5. Do: add up
164
the hundreds,
x 3
tens and
ones,
12
referring to
180
the grid and
the vertical
+300
format.
492
Record the
answer on
the vertical
format.
.
2 digit / 3 digit x 2 digit – grid method alongside recording vertical format – no apparatus
x
3
100
300
60
180
4
12
e.g. 43 x 24
1. Do: Multiply the
ones x ones
(3 x 4 = 12)
Record on the
grid, then ones x
tens (3 x 20)
Record on the
grid.
Model adding up
along the row to
record on the
vertical format.
Kate Mole Nonsuch Primary School
x
40
3
2014
20
4
60
12
43
x 24
72
Page 47
2. Do: Multiply the
tens x ones
(40 x 4 = 160)
Record on the
grid, then tens x
tens (40 x 20)
Record on the
grid.
Model adding up
along the row to
record on the
vertical format.
x
40
3
20
800
60
4
160
12
Add
3. Do: Use column
addition to add
up on the vertical
format.
43
x 24
72
+ 960
43
x 24
72
+ 960
1032
1
2 digit / 3 digit x 1 digit – short multiplication – formal written method
Begin with no carrying.
Note that in the KS2 National Tests, marks will not be awarded for use of the grid
method. Pupils MUST use the formal written methods of short and long multiplication to
be awarded the mark.
Kate Mole Nonsuch Primary School
2014
Page 48
2 digit / 3 digit x 2 digit – long multiplication – formal written method
Note where numbers are carried.
Begin with multiplying the unit
with each of the digits.
Children need to be taught
that the 0 in the second row is
written as a placeholder
because we are now
multiplying the tens with each
digit.
Kate Mole Nonsuch Primary School
2014
Page 49
Year 3, 4, 5 and 6 Ratio
How much flour will she need?
Kate Mole Nonsuch Primary School
2014
Page 50
Division
Mental Skills
Recognise the size and position of numbers
Count back in different steps 2s, 5s, 10s
Halve numbers to 20
Recognise division as repeated subtraction
Quick recall of division facts
Use known facts to derive associated facts
Divide by 10, 100, 1000 and understanding the effect
Divide by multiples of 10
Models and Images
Counting apparatus
Arrays
100 squares
Number Tracks
Numbered number lines
Marked but unnumbered numberlines
Empty numberlines
Multiplication squares
Number rods
Models and Images charts
ITPs – Multiplication Remainders Grid, Number Dials, Grouping
Bar Model for Division: 72 ÷ 8 =
Kate Mole Nonsuch Primary School
2014
Page 51
Year 1
Counting Choir
Count in steps of 2s, 5s and 10s forwards and
back from 0 and from any of its multiples using
the 100 Square and taking the opportunity to
discuss patterns that are recognized .
Half of 12.
Share dots equally one by one
Half of 6 = 3
½ of 6 = 3
1. Children learn to share objects practically.
6 muffins shared between 3
people is 2 each
6÷3=2
2. Children use grouping to solve problems involving division. With the help of
laminated sheets children place the given number of objects into groups using
the correct mathematical vocabulary. Please note: to distinguish between
grouping using multiplication and division a different type of grouping sheet is
used as shown below.
10 cookies into groups of 2.
How many groups?
10 ÷2 = 5
I have 10 cookies. Put them
into groups of 2. How many
groups have we got
altogether?
Kate Mole Nonsuch Primary School
2014
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3. Grouping with the use of jottings. Children first draw the total number of items
using dots and then put circles round the given number of dots. They count the
number of groups to obtain the answer.
I have 12 multilink cubes. If I
put them into groups of 2, how
many groups have I got?
Year 2
Counting Choir
Count in steps of 2s, 3s, 5s, 10s and
20s forwards and back from 0 and from
any of their multiples using the 100 /
200 square. Take the opportunity to
discuss patterns that are recognized.
Use known multiplication facts to work
out corresponding division facts
Know doubles and corresponding
halves. Extend to partitioning numbers,
then double / partitioning numbers then
halve.
Kate Mole Nonsuch Primary School
2014
Page 53
1. Children continue to use concrete materials and physical resources to share or
group objects equally with or without the help of laminated grouping sheets.
Sharing model of division:
Grouping model of division
12 muffins shared between 3
people is 4 each
12 ÷ 3 = 4
I have 18 strawberries. Put them
into groups of 3. How many have
we got altogether?
2. Grouping with the use of jottings. Children first draw the total number of items
using dots, then put circles around the given number of dots. They count the
number of groups to obtain the answer. In Y2 children are exposed to grouping in
all multiples between 2 and 9.
I have 18 multilink cubes. If I put
them into groups of 3, how many
groups have I got?
3. Children begin to use a marked numberline and number rods to solve division
problems. Always teach children to count on (not back) in groups.
18 ÷ 3 =
Kate Mole Nonsuch Primary School
2014
Page 54
Children then move on to just using a marked numberline and counting on in groups
This method requires children to
find out how many jumps of 3
they can make between 0 and
18. They circle 0 and 18 on the
numberline before they
commence their equal jumps of
3. The constant reinforcement of
vocabulary ‘into groups of’ is
very important.
4. More confident children, who are reliably able to count in multiples of 2,3,5 and
10, use an empty numberline to make their jumps.
Children write their own numbers
underneath the numberline each
time they complete a jump to
keep track of where they are.
The challenge in this process is
to remember to stop once they
get to the required number, in
this case, 18.
5. Reinforce division as grouping through arrays, using number rods and jottings,
and introduce remainders.
Kate Mole Nonsuch Primary School
2014
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6. Children begin to use a marked numberline to solve division problems involving
remainders.
The challenge in this process is to remember to not carry on jumping in multiples of 5
after number 25 and to realise that the remaining jumps need to be made in jumps of
ones to work out the remainder. Children must be able to count in multiples of 2, 3, 5
and 10 securely in order to use this method successfully.
Kate Mole Nonsuch Primary School
2014
Page 56
Years 3, 4, 5 and 6 Short Division HTO ÷ O
Children must have secure division facts knowledge to 10 x 10 in order for them to
see the benefits of this quick efficient method. Those children who are not yet ready
for this method should carry on with grouping through the use of arrays, numberlines
and other models as demonstrated in Year 2 policy.
The vocabulary may seem very advanced but it is important that it is modelled from
the beginning and subsequently throughout all KS2 year groups:
Divisor – the number that is being divided by; e.g. in the calculation 45 ÷ 5, 5 is the
divisor .
Dividend – the number that the divisor is being divided into. 45 ÷ 545 is the dividend.
Quotient – the answer to a division calculation. 45 ÷5 = 9. 9 is the quotient.
As children have to get used to a new layout which does not necessarily provide
conceptual understanding, it is imperative that the short division method is taught
using manipulatives (PV counters recommended) which shows the conceptual
understanding alongside the procedural method.
Click on the link below to view videos from the NCETM which very clearly
demonstrate how to move to a written method of division using manipulatives which
will support conceptual understanding.
https://www.ncetm.org.uk/resources/43589
1. Moving to a written method of division (short division)
2. Representing division with place value counters
3. Using place value counters and recording division
4. Division with remainders
5. Division with regrouping (labelled as exchange on the video title)
Kate Mole Nonsuch Primary School
2014
Page 57
Years 3, 4, 5 and 6 Long Division HTO ÷ TO
Chunking method – note that this method is not accepted as an efficient
written method in the KS2 national tests but is an important step in the
process
When children first learn to
divide by a 2 digit number
using chunking, provide them
with a fact box. As they
become more confident, they
can create their own fact box if
they wish. Answers are
expressed as whole number
remainders.
1
Fact Box
2 x 15 = 30
3 x 15 = 45
4 x 15 = 60
5 x 15 = 75
6 x 16 = 90
7 x 15 = 105
8 x 15 = 120
9 x 15 = 135
10 x 15 = 150
20 x 15 = 300
In Year 6 Children must also
be taught to express the
remainder as a fraction.
Kate Mole Nonsuch Primary School
2014
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In Year 6 Children will also be
taught the traditional ‘long
division’ written method. The
remainder can be expressed
as a decimal (remainder or
fraction). To express
remainders as a decimal, we
must carry on bringing down a
zero until we have no
remainders.
Note that children in Y6 must
use the traditional long division
(or short division) method to
gain the mark in the KS2
national tests – chunking will
not be awarded a mark.
Kate Mole Nonsuch Primary School
2014
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