Water Mill
Calculations
Policy
This is a guide to how calculations are recorded in different year groups
but there will be children working at different levels within each year group.
[1]
Addition
Y2
Y1
+ = signs and missing numbers
Children need to understand the concept of
equality before using the ‘=’ sign. Calculations
should be written either side of the equality sign so
that the sign is not just interpreted as ‘the answer’.
2 =1+ 1
2+3=4+1
3=3
2+2+2=4+2
+ = signs and missing numbers
Continue using a range of equations as in Year 1
but with appropriate, larger numbers.
Extend to
14 + 5 = 10 +
and
32 +
+ = 100 35 = 1 + + 5
Count on in tens and ones
23 + 12 =
Y3
+ = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate, larger numbers.
Pencil and paper procedures
e.g. 83 + 42 = 125
Horizontal expansion
80 + 3
+ 40 + 2
120 + 5 = 125
Missing numbers need to be placed in all
possible places.
Moving to column recording
1.
3+4=
3+ =7
+4=7
+ =7
300 + 60 + 7
+100 + 80 + 5
400 +140+12= 552
=3+4
7= +4
7=3+
7= +
2.
Activities
Children should have access to a wide range of
counting equipment, everyday objects, as well as
hoops, sorting trays, number tracks, numbered
number lines and digit and symbol cards.
Vertical expansion:
367
+185
12
140
400
552
Teacher modelling
Drawing jumps on numbered number lines to
support understanding of the mental method
3.
367
+185
552
11
Children
To create their own jumps using rulers, fingers,
pens, bodies etc.
7+4
[2]
Compact column:
Y4
+ = signs and missing numbers
Continue using a range of equations as in
Year 1 and 2 but with appropriate numbers.
Pencil and paper procedures up to 4 digits
367 + 185 = 552
1.
Building from Y 3:
300 + 60 + 7
+100 + 80 + 5
400 +140+12= 552
2.
Vertical expansion:
367
+185
12
140
400
552
3.
Addition
Y5
+ = signs and missing numbers
Continue using a range of equations as in Year
1 and 2 but with appropriate numbers.
Pencil and paper procedures
Extend to numbers with at least four digits then
more than 4
3587 + 675 = 4262
3587
+ 675
4262
111
Pencil and paper procedures
Extend to numbers with any number of digits
and decimals with 1, 2 and/or 3 decimal
places.
13.86 + 9.481 = 23.341
13.86
+ 9.481
23.341
11 1
Revert to expanded methods if the children
experience any difficulty.
Compact column:
Y6
+ = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate numbers.
Extend to up to two places of decimals (same
number of decimals places) and adding several
numbers (with different numbers of digits).
72.8
+54.6
127.4
367
+185
552
11
Extend to decimals in the context of money.
[3]
Revert to expanded methods if the children
experience any difficulty.
Y1
Subtraction
Y2
Understand subtraction as 'take away'
Find a small difference by counting up
42 – 39 = 3
Calculate by counting back;
I have 11 toy cars. There are 5 cars too many to fit
in the garage. How many cars fit in the garage?
11-6=5
Subtract 9 or 11. Begin to add/subtract 19 or 21
35 – 9 = 26
Use practical and informal written methods to
support the subtraction of a one-digit number from
a one digit or two-digit number and a multiple of 10
from a two-digit number.
Use the vocabulary related to addition and
subtraction and symbols to describe and record
addition and subtraction number sentences
Recording by
- drawing jumps on prepared lines
- constructing own lines
Use known number facts and place value to
subtract
Y3
Subtract mentally a ‘near multiple of 10’ to or from a
two-digit number
Continue as in Year 2 but with appropriate numbers
e.g. 78 – 49 is the same as 78 -50 + 1
Use known number facts and place value to subtract
Continue as in Year 2 but with appropriate numbers
e.g. 97 – 15 = 72
With practice, children will need to record less
information and decide whether to count back or
forward. It is useful to ask children whether
counting up or back is the more efficient for
calculations such as 57 – 12, 86 – 77 or 43 – 28
Pencil and paper procedures
Partition subtraction
Bridge through 10 where necessary
32-17
Moving towards
- = signs and missing numbers
7-3=
=7-3
7- =4
4= -3
-3=4
4=7- =4
4= - = signs and missing numbers
Continue using a range of equations as in Year 1 but
with appropriate numbers.
Extend to 14 + 5 = 20 - _
[4]
- = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate numbers.
Y4
Find a small difference by counting back
e.g. 5003 – 4996 = 7
This can be modelled on an empty number line.
Children should be encouraged to use known
number facts to reduce the number of steps.
Subtract the nearest multiple of 10, then
adjust.
Continue as in Year 2 and 3 but with appropriate
numbers.
Use known number facts and place value to
subtract
92-25=67
Subtraction
Y5
Find a difference by counting back
e.g. 8006 – 2993 = 5013
This can be modelled on an empty number
line (see complementary addition below).
Subtract the nearest multiple of 10 or 100, then
adjust.
Continue as in Year 2, 3 and 4 but with
appropriate numbers.
Use known number facts and place value to
subtract
6.1 – 2.4 = 3.7
Y6
Find a difference by counting back
e.g. 8000 – 2785 = 5215
To make this method more efficient, the
number of steps should be reduced to a
minimum through children knowing:
Complements to 1, involving decimals to two
decimal places ( 0.16 +
0.84)
Complements to 10, 100 and 100
Subtract the nearest multiple of 10, 100 or
1000, then adjust
Continue as in Year 2, 3, 4 and 5 but with
appropriate numbers.
Use known number facts and place
value to subtract
0.5 – 0.31 =0.19
Use column method recording up to 4 digits
- = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate numbers.
- = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate numbers.
Reduce the number of steps to make the
calculation more efficient. Extend to 2 places of
decimals.
- = signs and missing numbers
Continue using a range of equations as in Year 1
and 2 but with appropriate numbers.
[5]
Multiplication
Y2
Y1
Multiplication is related to doubling and counting
groups of the same size.
Looking at columns
2+2+2
3 groups of 2
Looking at rows
3+3
2 groups of 3
Counting using a variety of practical resources
Counting in 2s e.g. counting socks, shoes, animal’s
legs…
Counting in 5s e.g. counting fingers, fingers in
gloves, toes…
Counting in 10s e.g. fingers, toes…
x = signs and missing numbers
7x2=
=2x7
7 x = 14
14 = x 7
x 2 = 14
14 = 2 x
x = 14
14 = x
Arrays and repeated addition
4 x 2 or 4 + 4
2 x 4 or 2 + 2 + 2 + 2
Y3
x = signs and missing numbers
Continue using a range of equations as in Year 2
but with appropriate numbers.
Arrays and repeated addition
Continue to understand multiplication as
repeated addition and continue to use arrays
(as in Year 2).
Doubling multiples of 5 up to 50
35 x 2 = 70
Doubling multiples of 5 up to 50
15 x 2 = 30
Partition
Children need to be secure with partitioning
numbers into 10s and 1s and partitioning in different
ways: 6 = 5 + 1 so e.g. Double 6 is the same as
double five add double one.
Pictures / marks
There are 3 sweets in one bag.
How many sweets are there in 5 bags?
[6]
Use known facts and place value to carry out
simple multiplications
Use the same method as above (partitioning), e.g.
Multiplication
Y5
Y4
Y6
x = signs and missing numbers
Continue using a range of equations as in Year
2 but with appropriate numbers
x = signs and missing numbers
Continue using a range of equations as in Year
2 but with appropriate numbers
x = signs and missing numbers
Continue using a range of equations as in Year
2 but with appropriate numbers
Partition
Use the grid method of multiplication (as below)
Partition
Use the grid method of multiplication (as below)
Partition
Use the grid method of multiplication (as below)
Pencil and paper procedures
Grid method
72 x 38 is approximately 70 x 40 = 2800
Pencil and paper procedures
Grid method
372 x 24 is approximately 400 x 20 = 8000
Pencil and paper procedures
Grid method
23 x 7 is approximately 20 x 10 = 200
x
7
20
140
3
21
= 161
Move towards Short multiplication method
346
X 9
2700 300 x 9
360 40 x 9
54 6 x 9
3114
11
Leading to
346
X 9
3114
11
Extend to simple decimals with one decimal place.
Short Multiplication up to 4 digits by 1 digits
htu x u
Leading to
346
X 9
3114
11
346
X 9
2700 300 x 9
360 40 x 9
54 6 x 9
3114
11
Extend to decimals with up to two decimal places.
Short Multiplication
use larger numbers thtu x u
346
X 9
3114
11
Long Multiplication
72
X 38
2160 72 x 30
576 72 x 8
2736
1
Long Multiplication
72
X 38
2160 72 x 30
576 72 x 8
2736
1
[7]
Division
Y2
Y1
Sharing
Requires secure counting skills
-see counting and understanding number strand
Develops importance of one-to-one
correspondence See appendix for additional
information on x and ÷ and aspects of number
Sharing – 6 sweets are shared between 2 people.
How many do they have each?


Practical activities involving sharing, distributing
cards when playing a game, putting objects onto
plates, into cups, hoops etc.
Grouping
Sorting objects into 2s / 3s/ 4s etc
How many pairs of socks are there?
There are 12 crocus bulbs. Plant 3 in each pot.
How many pots are there?
Jo has 12 Lego wheels. How many cars can
she make?
÷ = signs and missing numbers
6÷2=
=6÷2
6÷ =3
3=6 ÷
÷2=3
3= ÷2
÷ =3
3= ÷
Grouping
Link to counting and understanding number strand
Count up to 100 objects by grouping them and
counting in tens, fives or twos;…
Find one half, one quarter and three quarters of
shapes
and sets of objects
6 2 can be modelled as:
There are 6 strawberries.
How many people can have 2 each? How many
2s make 6?
6
2 can be modelled as:
In the context of money count forwards and
backwards using 2p, 5p and 10p coins
Practical grouping e.g. in PE
12 children get into teams of 4 to play a game. How
many teams are there?
[8]
Y3
÷ = signs and missing numbers
Continue using a range of equations as in Year 2
but with appropriate numbers.
Understand division as sharing and grouping
18 ÷ 3 can be modelled as:
Sharing – 18 shared between 3 (see Year 1
diagram)
OR
Grouping - How many 3’s make 18?
Remainders
16 ÷ 3 = 5 r1
Sharing - 16 shared between 3, how many left over?
Grouping – How many 3’s make 16, how many left
over?
e.g.
Y4
÷ = signs and missing numbers
Continue using a range of equations as in Year 2
but with appropriate numbers.
Sharing and grouping
30 ÷ 6 can be modelled as:
grouping – groups of 6 placed on no. line and the
number
of groups counted e.g.
sharing – sharing among 6, the number given
to each person
Pencil and paper procedures
Remainders
41 ÷ 4 = 10 r1
Division
Y5
÷ = signs and missing numbers
Continue using a range of equations as in Year
2 but with appropriate numbers.
Sharing and grouping
Continue to understand division as both sharing
and grouping (repeated subtraction).
Remainders
Quotients expressed as fractions or decimal
fractions
61 ÷ 4 = 15 ¼ or 15.25
Remainders
Quotients expressed as fractions or decimal
fractions 676 ÷ 8 = 84.5
Pencil and paper procedures
72 ÷ 5 lies between 50 5 = 10 and 100 5 = 20
Partition the dividend into multiples of the
divisor:
e.g.
72
50 (10 groups)
22
- 20 (4 groups)
2
Answer : 14 remainder 2
Move towards Bus shelter method then short
division
Pencil and paper procedures
72 ÷ 5 lies between 50 5 = 10 and 100 5 = 20
Partition the dividend into multiples of the
divisor:
e.g.

-
72
50
22
20
2
Y6
÷ = signs and missing numbers
Continue using a range of equations as in Year
2 but with appropriate numbers.
Sharing and grouping
Continue to understand division as both
sharing and grouping (repeated subtraction).
(10 groups)
(4 groups)
Answer : 14 remainder 2
[9]
Pencil and paper procedures
977 ÷ 36 is approximately 1000
40 = 25
Partition the dividend into multiples of the
divisor:
e.g.
OR
977
720 (20 groups)
257
- 180 (5 groups)
77
72 (2 groups)
5
Answer: 27 5/36
Reception
Place value, estimation, ordering and counting
Year 1
Counting
• Say and use the number names in order In
familiar contexts such as number rhymes, songs,
stories, counting games and activities (first to five,
then ten, then twenty and beyond).
• Recite the number names in order, continuing the
count forwards or backwards from a given number.
• Count reliably up to 10 everyday objects (first to
5, then 10, then beyond), giving just one number
name to each object.
Recognise small numbers without counting.
• Begin to recognise ‘none’ and ‘zero’ in stories,
rhymes and when counting.
• Count reliably in other contexts, such as clapping
sounds or hopping movements.
• Count in tens.
• Count in twos.
• Estimate a number in the range that can be
counted reliably, then check by counting.
Comparing and ordering numbers
• Use language such as more or less, greater or
smaller, to compare two numbers and say which
is more or less, and say a number which lies
between two given numbers.
• Order a given set of numbers: for example, the set
of numbers 1 to 6 given in random order.
• Order a given set of selected numbers:
for example, the set 2, 5, 1, 8, 4.
• Begin to understand and use ordinal numbers in
different contexts.
Counting, properties of numbers and number
sequences
• Know the number names and recite them in order
to at least 100, from and back to zero.
• Count reliably at least 20 objects.
• Describe and extend number sequences:
count on and back in ones from any small
number, and in tens from and back to zero;
count on in twos from zero, then one, and begin to
recognize odd or even numbers to about 100 as
‘every other number’;
count in steps of 5 from zero to100
Place value and ordering
Read and write numerals from 0 to at least 20.
• Begin to know what each digit in a two-digit
number represents. Partition a ‘teens’ number and
begin to partition larger two-digit numbers into a
multiple of 10 and ones (TU).
• Understand and use the vocabulary of
comparing and ordering numbers, including
ordinal numbers to at least 20.
Use the = sign to represent equality.
Compare two familiar numbers, say which is more
or less, and give a number which lies between them.
• Within the range 0 to 30, say the number that is
1 or 10 more or less than any given number.
• Order numbers to at least 20, and position them
on a number track.
Estimating
• Understand and use the vocabulary of estimation.
Give a sensible estimate of a number of objects that
can be checked by counting (e.g. up to about 30
objects).
Year 2
Counting, properties of numbers and number
sequences
• Say the number names in order to at least 100,
from and back to zero.
• Count reliably up to 100 objects by grouping them:
for example, in tens, then in fives or twos.
• Describe and extend simple number
sequences: count on or back in ones or tens,
starting from any two digit number;
count in hundreds from and back to zero; count on
in twos from and back to zero or any small number,
and recognise odd and even numbers to at least
30; count on in steps of 3, 4 or 5 to at least 30, from
and back to zero, then from and back to any given
small number.
• Begin to recognise two-digit multiples of 2, 5 or 10.
Place value and ordering
• Read and write whole numbers to at least 100
in figures and words.
• Know what each digit in a two-digit number
represents, including 0 as a place holder, and
partition two-digit numbers into a multiple of ten and
ones (TU).
• Use and begin to read the vocabulary of
comparing and ordering numbers, including ordinal
numbers to 100. Use < > = signs
Use the = sign to represent equality.
Compare two given two-digit numbers, say which is
more or less, and give a number which lies between
them.
• Say the number that is 1 or 10 more or less than
any given two-digit number.
• Order whole numbers to at least 100, and
position them on a number line and 100 square.
Estimating and rounding
• Use and begin to read the vocabulary of estimation
and approximation; give a sensible estimate of at
least 50 objects.
[10]
• Round numbers less than 100 to the nearest 10.
Place value, estimation, ordering and counting
Year 3
Year 4
Counting, properties of numbers and number sequences
• Count larger collections by grouping them: for example, in tens, then other
numbers.
• Describe and extend number sequences:
count on or back in tens or hundreds, starting from any two- or three-digit
number.
count on or back in twos starting from any two-digit number, and recognise odd
and even numbers to at least 100;
count on in steps of 4 or 8, 50 and 100
• Recognise two-digit and three-digit multiples of 4, 8 and three-digit multiples of
50 and 100.
Place value and ordering
• Read and write whole numbers to at least 1000 in figures and words.
• Know what each digit represents, and partition three-digit numbers into a
multiple of 100, a multiple of ten and ones (HTU).
• Read and begin to write the vocabulary of comparing and ordering numbers,
including ordinal numbers to at least 100.
Compare two given three-digit numbers, say which is more or less, and give a
number which lies between them.
• Say the number that is 1, 10 or 100 more or less than any given two- or threedigit number.
• Order whole numbers to at least 1000, and position them on a number line.
Estimating and rounding
• Read and begin to write the vocabulary of estimation and approximation.
Give a sensible estimate of up to about 100 objects.
• Round any two-digit number to the nearest 10 and any three digit number to
the nearest 100.
Place value, ordering and rounding (whole numbers)
• Read and write whole numbers to at least 10 000 in figures and words, and
know what each digit represents.
Partition numbers into thousands, hundreds, tens and ones.
• Add/subtract 1, 10, 100 or 1000 to/from any integer, and count on or back in
tens, hundreds or thousands from any whole number up to 10000.
• Multiply or divide any integer up to 1000 by 10 (whole-number answers), and
understand the effect. Begin to multiply by 100.
• Read and write the vocabulary of comparing and ordering numbers.
Use symbols correctly, including less than (<), greater than (>), equals (=).
Give one or more numbers lying between two given numbers and order a set of
whole numbers less than 10000.
• Read and write the vocabulary of estimation and approximation. Make and
justify estimates up to about 250, and estimate a proportion.
Round any positive integer less than 1000 to the nearest 10 or 100.
• Recognise negative numbers in context (e.g. on a number line, on a
temperature scale).
Read Roman numerals to 100
Properties of numbers and number sequences
• Recognise and extend number sequences formed by counting from any
number in steps of constant size, extending beyond zero when counting back:
for example, count on in steps of 25 to 500, and then back to, say, –100.
• Recognise odd and even numbers up to 1000, and some of their properties,
including the outcome of sums or differences of pairs of odd/even numbers.
• Recognise multiples of 6, 7, 9, 25 and 1000
[11]
Place value, estimation, ordering and counting
Year 5
Year 6
Place value, ordering and rounding
• Read and write whole numbers in figures and words, and know what each
digit represents. (decimals) up to 1 000 000
• Multiply and divide any positive integer up to 10000 by 10 or 100 and
understand the effect (e.g. 9900 ÷ 10, 737 ÷ 10, 2060 ÷ 100).
• Use the vocabulary of comparing and ordering numbers, including symbols
such as <, >, =. Give one or more numbers lying between two given numbers.
Order a set of integers less than 1 million. (decimals)
• Use the vocabulary of estimation and approximation.
Make and justify estimates of large numbers, and estimate simple proportions
such as one third, seven tenths.
Round any integer up to 10000 to the nearest 10, 100 or 1 000, 10 000,
100 000(rounding decimals)
• Order a given set of positive and negative integers (e.g. on a number line,
on a temperature scale). Calculate a temperature rise or fall across 0 °C.
Read Roman numerals up to 1000
Place value, ordering and rounding
• Multiply and divide decimals mentally by 10 or 100, and integers by 1000,
and explain the effect.
• Use the vocabulary of estimation and approximation.
Numbers up to 100 000
Consolidate rounding an integer to the nearest 10, 100 or 1000.
• Find the difference between a positive and a negative integer, or two negative
integers, in a context such as temperature or the number line, and order a set
of positive and negative integers.
Properties of numbers and number sequences
• Recognise and extend number sequences formed by counting from any
number in steps of constant size, extending beyond zero when counting back.
For example:
count on in steps of 25 to 1000, and then back;
count on or back in steps of 0.1, 0.2, 0.3…
• Make general statements about odd or even numbers, including the outcome
of sums and differences.
• Recognise multiples of 6, 7, 8, 9, up to the 10th multiple.
Know and apply tests of divisibility by 2, 4, 5, 10 or 100.
• Know squares of numbers to at least 10 10.
• Find all the pairs of factors of any number up to 100.
Properties of numbers and number sequences
• Recognise and extend number sequences, such as the sequence of square
numbers, or the sequence of triangular numbers 1, 3, 6, 10, 15…
Count on in steps of 0.1, 0.2, 0.25, 0.5…, and then back.
• Make general statements about odd or even numbers, including the outcome
of products.
• Recognise multiples up to 10 10. Know and apply simple tests of divisibility.
Find simple common multiples.
• Recognise squares of numbers to at least 12 12.
• Recognise prime numbers to at least 20. Factorise numbers to 100 into prime
factors.
[12]