Written Calculation Policy Key Stage 1 and 2

Written Calculation Policy Key Stage 1 and 2
Written Calculation Policy
Key Stage 1 and 2
Review date: January 2013
Kibworth Primary School
Pencil and paper procedures
Background to the policy
This policy contains the key pencil and paper procedures that will be taught within our school. It has been written to ensure
consistency and progression throughout the school and reflects a whole school agreement.
Although the focus of the policy is on pencil and paper procedures it is important to recognise that the ability to calculate
mentally lies at the heart of the Primary National Strategy for mathematics. The mental methods in the Primary
Framework for teaching mathematics will be taught systematically from Reception onwards and pupils will be given
regular opportunities to develop the necessary skills. However mental calculation is not at the exclusion of written
recording and should be seen as complementary to and not as separate from it. In every written method there is an
element of mental processing. Sharing written methods with the teacher encourages children to think about the mental
strategies that underpin them and to develop new ideas. Therefore written recording both helps children to clarify their
thinking and supports and extends the development of more fluent and sophisticated mental strategies.
During their time at this school children will be encouraged to see mathematics as both a written and spoken language.
Teachers will support and guide children through the following important stages:
• developing the use of pictures and a mixture of words and symbols to represent numerical activities;
• using standard symbols and conventions;
• use of jottings to aid a mental strategy;
• use of pencil and paper procedures;
• use of expanded and compact calculation methods
• use of a calculator.
This policy concentrates on the introduction of standard symbols, the use of the empty number line as a jotting to aid mental calculation
and on the introduction of expanded and compact calculation methods. It is important that children do not abandon jottings and mental
methods once pencil and paper procedures are introduced. Children will always be encouraged to look at a calculation/problem and then
decide which is the best method to choose – pictures, mental calculation with or without jottings, structured recording or a calculator. Our
long-term aim is for children to be able to select an efficient method of their choice (whether this be mental, written or using a calculator)
that is appropriate for a given task. They will do this by always asking themselves:
‘Can I do this in my head?’
‘Can I use drawings or jottings to support my mental calculation?
‘Do I need to use an expanded or compact calculation method?’
‘Do I need a calculator?’
The policy is written in stages that demonstrate a clear progression towards the development of formal written algorithm.
Children will only progress on to the next stage for each operation when they have mastered the pencil and paper procedures in
their current stage.
Addition
Stage 1
Stage 2
Stage 3
Addition Objectives
Addition Objectives
Addition Objectives
PF-calculating
PF-calculating
PF-calculating
Relate addition to counting on; recognise that
Add or subtract mentally a one-digit number or a
Develop and use written methods to record,
addition can be done in any order; use practical
multiple of 10 to or from any two-digit number; use
support or explain addition and subtraction of two-
and informal written methods to support the
practical and informal written methods to add and
digit and three-digit numbers
addition of a one-digit number or a multiple of 10
subtract two-digit numbers
to a one-digit or two-digit number
Add or subtract mentally combinations of one-digit
Understand that subtraction is the inverse of
Use the vocabulary related to addition and
addition and vice versa; use this to derive and
subtraction and symbols to describe and record
record related addition and subtraction number
addition and subtraction number sentences
sentences
Use the symbols
, -,
,
and
and two-digit numbers
to record and
interpret number sentences involving all four
operations; calculate the value of an unknown
in a number sentence (e.g.
PF-knowing and using number facts
2
6, 30 -
24)
Derive and recall all pairs of numbers with a total of
10 and addition facts for totals to at least 5; work
out the corresponding subtraction facts
PF-knowing and using number facts
PF-knowing and using number facts
Derive and recall all addition and subtraction facts
Derive and recall all addition and subtraction facts
for each number to at least 10, all pairs with totals
for each number to 20, sums and differences of
to 20 and all pairs of multiples of 10 with totals up
multiples of 10 and number pairs that total 100
to 100
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental
strategies, mental with jottings, calculator or written procedures.
Addition
Stage 1
Stage 2
+ = 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
Missing numbers need to be placed in all possible places.
3+4=
3+ =7
+4=7
+∇=7
=3+4
7= +4
7=3+
7= +∇
+ = 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
Partition in different ways and recombine
12 + 23 = 10 + 2 + 20 + 3
= 30 + 5
= 35
Count on in tens and ones
23 + 12 = 23 + 10 + 2
= 33 + 2
= 35
+10
Teacher modelling
Drawing jumps on numbered number lines to support
understanding of the mental method
33
7+ 4
0
1
2
3
4
5
6
7
8
9
10
11
Partition into tens and ones
•
Partition both numbers and recombine.
•
Count on by partitioning the second number only
e.g.
36 + 53 = 53 + 30 + 6
= 83 + 6
= 89
+6
83
53
89
Add a near multiple of 10 to a two-digit number
Secure mental methods by using a number line to
model the method.
Continue as in Year 2 but with appropriate numbers
e.g. 35 + 19 is the same as 35 + 20 – 1.
35
Partitioning and bridging through 10.
The steps in addition often bridge through a multiple of 10
e.g.
Children need to be secure adding multiples of 10 to any
two-digit number including those that are not multiples of
10.
48 + 36 = 84
Children should be able to partition the 7 to relate adding the
2 and then the 5.
+30
+2
+4
8 + 7 = 15
48
8
Children
To create their own jumps using rulers, fingers, pens,
bodies etc.
+ = signs and missing numbers
Continue using a range of equations as in Year 1 and 2
but with appropriate, larger numbers.
+30
+2
23
Activities
Children should have access to a wide range of counting
equipment, everyday objects, as well as hoops, sorting
trays, number tracks and numbered number lines.
Stage 3
10
+5
84
Expanded written method
83 + 42 = 125
15
+2
Add 9 or 11 by adding 10 and adjusting by 1
e.g.
Add 9 by adding 10 and adjusting by 1
35 + 9 = 44
+10
12
35
78 80
44
45
-1
REMEMBER: Children should have the opportunity to choose the appropriate
strategies, mental with jottings, calculator or written procedures.
either
or
1. Vertical expansion
2. Horizontal expansion
83
80 + 3
+ _42
+ 40 + 2
5
120 + 5 = 125
120
125
Children should be able to make the choice of
strategy
whether
this
is, mental
reverting to
the expanded
method
if experiencing
difficulty.
Addition
Stage 4
Addition Objectives
PF-calculating
Refine and use efficient written methods to add and
subtract two-digit and three-digit whole numbers
and .p
Stage 6
Addition Objectives
Addition Objectives
PF-calculating
PF-calculating
Use efficient written methods to add and subtract
Use efficient written methods to add and subtract
whole numbers and decimals with up to two places
integers and decimals, to multiply and divide
integers and decimals by a one-digit integer, and
to multiply two-digit and three-digit integers by a
Add or subtract mentally pairs of two-digit whole
numbers (e.g. 47
Stage 5
two-digit integer
58, 91 - 35)
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies, mental with jottings,
calculator or written procedures.
Addition
Stage 4
+ = signs and missing numbers
Continue using a range of equations as in Stage 1 and 2
but with appropriate numbers.
Add the nearest multiple of 10, then adjust
Continue as in Stage 2 and 3 but with appropriate numbers
e.g. 63 + 29 is the same as 63 + 30 - 1
Towards a compact written method
367 + 185 = 431
either
or
367
+185
12
140
400
552
300 + 60 + 7
100 + 80 + 5
400 +140+12 = 552
leading to
367
+185
552
Stage 5
Stage 6
+ = signs and missing numbers
Continue using a range of equations as in Stage 1 and 2
but with appropriate numbers.
+ = signs and missing numbers
Continue using a range of equations as in Stages 1 and 2
but with appropriate numbers.
Add or subtract the nearest multiple of 10 or 100, then
adjust
Continue as in Stage 2, 3 and 4 but with appropriate
numbers e.g. 458 + 79 = is the same as 458 + 80 - 1
Add the nearest multiple of 10, 100 or 1000, then
adjust
Continue as in Stages 2, 3, 4 and 5 but with appropriate
numbers including extending to adding 0.9, 1.9, 2.9 etc.
Compact written method
Extend to numbers with at least four digits
3587 + 675 = 4262
Compact written method
Extend to numbers with any number of digits and
decimals with 1, 2 and/or 3 decimal places.
13.86 + 9.481 = 23.341
3587
+ 675
4262
111
Children should be able to make the choice of
reverting to expanded methods if experiencing any
difficulty.
Extend to up to two places of decimals (same number of
decimals places) and adding several numbers (with
different numbers of digits).
13.86
+ 9.481
23.341
11 1
Children should be able to make the choice of
reverting to expanded methods if experiencing any
difficulty.
100 10
Extend to decimals in the context of money.
72.8
+ 54.6
127.4
1 1
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies, mental with jottings,
calculator or written procedures.
Subtraction
Year 1
Subtraction Objectives
Year 2
Year 3
Subtraction Objectives
Subtraction Objectives
PF-calculating
PF-calculating
Understand subtraction as 'take away' and find a
Add or subtract mentally a one-digit number or a
'difference' by counting up; use practical and
multiple of 10 to or from any two-digit number;
informal written methods to support the subtraction
use practical and informal written methods to add
of a one-digit number from a one digit or two-digit
and subtract two-digit numbers
PF-calculating
Develop and use written methods to record, support
or explain addition and subtraction of two-digit and
three-digit numbers
number and a multiple of 10 from a two-digit
number
Add or subtract mentally combinations of one-digit
Understand that subtraction is the inverse of
and two-digit numbers
addition and vice versa; use this to derive and
Use the vocabulary related to addition and
subtraction and symbols to describe and record
record related addition and subtraction number
sentences
addition and subtraction number sentences
Use the symbols
, -,
,
and
to record
and interpret number sentences involving all
PF-knowing and using number facts
four operations; calculate the value of an
Derive and recall all addition and subtraction facts
unknown in a number sentence (e.g.
6, 30 -
24)
2
for each number to 20, sums and differences of
multiples of 10 and number pairs that total 100
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Subtraction
Stage 1
Stage 2
Stage 3
Children need to be given experience of all the strategies in a range of contexts, in order for them to make an informed choice of
the most appropriate method to use when completing calculations.
- = 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 Stage 1 but with
appropriate numbers.
Extend to 14 + 5 = 20 Find a small difference by counting up
42 – 39 = 3
Understand subtraction as 'take away'
+1
•
39
Find a 'difference' by counting up;
I have saved 5p. The socks that I want to buy cost 11p.
How much more do I need in order to buy the socks?
+6
- = signs and missing numbers
Continue using a range of equations as in Stage 1 and 2
but with appropriate numbers.
Find a small difference by counting up
Continue as in Stage 2 but with appropriate numbers e.g.
102 – 97 = 5
+2
40
42
Subtract 9 or 11. Begin to add/subtract 19 or 21
35 – 9 = 26
Subtract mentally a ‘near multiple of 10’ to or from a
two-digit number
Continue as in Stage 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 Stage 2 but with appropriate numbers e.g.
97 – 15 = 72
82
+1
0
1 2 3
4
5
6
7 8
9
•
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.
I have 11 toy cars. There are 5 cars too many to fit in the
garage. How many cars fit in the garage?
-5
72
26
11
35
-10
97
27
-2
-5
-10
With practice,-5children will need-20
to record less
Use known number facts and place value to subtract
(partition second number only)
37 – 12 = 37 – 10 – 2
= 27 – 2
= 25
25
6
97
77
10 11 12
25
0
87
37
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.
Expanded written method without decomposition
168 - 54
-10
Use the vocabulary related to addition and subtraction and
Bridge through 10 where necessary
symbols to describe and record addition and subtraction
32 - 17
number sentences
32
15
20
22
Recording by
REMEMBER:
Children
- drawing
jumps on prepared
linesshould have the opportunity to choose the appropriate strategy
-5
-2
-10
- constructing
own lines
mental with
jottings, calculator or written procedures.
100 + 60 + 8
50 + 4
100
+
10
+ 4is,= mental
114
whether this
strategies,
Subtraction
Stage 4
Subtraction Objectives
Stage 5
Stage 6
Subtraction Objectives
Subtraction Objectives
Refine and use efficient written methods to add and
PF-calculating
PF-calculatting
subtract two-digit and three-digit whole numbers
Use efficient written methods to add and subtract
Use efficientt written methods to add and subtract
and
whole numbers and decimals with up to two places
integers and
d decimals, to multiply and divide
Extend mental-methods for whole-number
integers and
d decimals by a one-digit integer, and to
Add or subtract mentally pairs of two-digit whole
calculations, for example to subtract one near-
multiply two
o-digit and three-digit integers by a
numbers (e.g. 47
multiple of 1000 from another (e.g. 6070 - 4097)
two-digit intteger
PF-calculating
.p
58, 91 - 35)
Calculate me
entally with integers and decimals:
U.t
PF – Knowing & Using Number Facts
Use knowledge of addition and subtraction facts
and place value to derive sums and differences
of pairs of multiples of 10, 100 or 1000
U.t,
PF – Knowing & Using Number Facts
Use knowledge of place value and
addition and subtraction of two-digit
numbers to derive sums and differences
of decimals (e.g. 6.5 ± 2.7, half of 5.6, double
0.34)
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Subtraction
Stage 4
Stage 5
Stage 6
- = signs and missing numbers
Continue using a range of equations as in Stages 1 and 2
but with appropriate numbers.
- = signs and missing numbers
Continue using a range of equations as in Stages 1 and 2
but with appropriate numbers.
- = signs and missing numbers
Continue using a range of equations as in Stages 1 and 2
but with appropriate numbers.
Find a small difference by counting up
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.
Find a difference by counting up
e.g. 8006 – 2993 = 5013
This can be modelled on an empty number line.
Find a difference by counting up
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 Stages 2, 3, 4 and 5 but with appropriate
numbers.
Subtract the nearest multiple of 10, then adjust.
Continue as in Stages and 3 but with appropriate numbers.
Expanded written method (with and without
decomposition)
Subtract the nearest multiple of 10 or 100, then adjust.
Continue as in Stages 2, 3 and 4 but with appropriate
numbers.
Towards a compact written method with and without
decomposition)
Without decomposition
100 + 70 + 8
40 + 7
100 + 30 + 1 = 131
7658
-5 3 4 2
2316
Compact written method
Continue to use a range of equations as in Stage 5 but
with appropriate numbers.
4 12 13 10+
80 10+
100 + 90 + 2
50 + 7
100 + 30 + 5
With decomposition
410+
854
-247
607
Extend to decimals (hundredths).
10+
Extend to decimals (tenths).
5 10+
8.7
-2.4
6.3
5345
-3 4 6 8
1877
6.5
-1.7
4.8
12.42
- 4.31
8.11
Multiplication
=
Stage 1
Stage 2
Stage 3
Multiplication Objectives
Multiplication Objectives
Multiplication Objectives
PF-calculating
Solve practical problems that involve combining groups of
2, 5 or 10, or sharing into equal groups
PF-calculating
Represent repeated addition and arrays as multiplication,
and sharing and repeated subtraction (grouping) as
division; use practical and informal written methods and
related vocabulary to support multiplication and division,
including calculations with remainders
PF-calculatin
ng
Use practical and informal written methods to multiply and
divide two-digit numbers (e.g. 13 3, 50 4); round
remainders up
p or down, depending on the context
Multiply one-d
digit and two-digit numbers by 10 or 100, and
describe the effect
e
PF-knowing & using number facts
Count on or back in ones, twos, fives and tens and use this
knowledge to derive the multiples of 2, 5 and 10 to the
tenth multiple
PF-knowing & using number facts
Derive and recall multiplication facts for the 2, 5 and 10
times-tables and the related division facts; recognise
multiples of 2, 5 and 10
PF-knowing & using number facts
Derive and recall multiplication facts for the 2, 3, 4, 5, 6
and 10 times--tables and the corresponding division facts;
recognise multiples of 2, 5 or 10 up to 1000
REMEMBER: Children should have the opportunitty to choose the appropriate strategy whether this is, mental strattegies, mental with jottings,
calculator or written procedures.
Multiplication
Stage 2
Stage 1
Multiplication is related to doubling and counting groups of
the same size.
Stage 3
x = signs and missing numbers
7x2=
=2x7
7 x = 14
14 = x 7
x 2 = 14
14 = 2 x
x ∇ = 14
14 = x ∇
x = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers.
Arrays and repeated addition
Continue to understand multiplication as repeated addition
and continue to use arrays (as in Stage 2).
Arrays and repeated addition
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…
Pictures / marks
4 x 2 or 4 + 4
2 x 4 or 2 + 2 + 2 + 2
4 x 2 or 4 + 4
2 x 4 or 2 + 2 + 2 + 2
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
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.
There are 3 sweets in one bag.
How many sweets are there in 5 bags?
Doubling multiples of 5 up to 50
35 x 2 = 70
Partition
X
30
5
2
60
10
=70
AND double 15
10
+
5
x2
20
+
10 = 30
Use known facts and place value to carry out simple
multiplications
Use the same method as above (partitioning), e.g.
32 x 3 = 96
Leading to
x
30 2
3
90 6
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies, mental with jottings,
= 96
2 20
10
= 30
calculator or written procedures.
X
10
5
Multiplication
=
Stage 4
Stage 5
Stage 6
Multiplication Objectives
Multiplication Objectives
Multiplication Objectives
PF-calculating
PF-calculating
PF-calculatting
Develop and use written methods to record,
Refine and use efficient written methods to multiply
Use efficientt written methods to add and subtract
support and explain multiplication and division of
and divide HTU
integers and
d decimals, to multiply and divide
U, TU
TU, U.t
U and HTU
U
two-digit numbers by a one-digit number, including
integers and
d decimals by a one-digit integer, and to
division with remainders (e.g. 15
multiply two
o-digit and three-digit integers by a
9, 98
6)
two-digit intteger
Multiply and divide numbers to 1000 by 10 and
then 100 (whole-number answers), understanding
the effect; relate to scaling up or down
PF-Knowing and using number
PF-Knowing and using number
PF-Knowing and using number
facts
facts
facts
Derive and recall multiplication facts up to 10 x 10,
Recall quickly multiplication facts up to 10 x 10 and
Use knowled
dge of place value and multiplication
the corresponding division facts and multiples of
use them to multiply pairs of multiples of 10 and
numbers to 10 up to the tenth multiple
100; derive quickly corresponding division facts
facts to
10
10 to derive
d
related multiplication and division
facts involv
ving decimals (e.g. 0.8
7, 4.8
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
6)
Multiplication
Stage 4
Stage 5
x = signs and missing numbers
Continue using a range of equations as in Stage 2 but
with appropriate numbers
x = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers
Expanded grid method
Extend to multiplying by 7, 8 and 9
Expanded grid method
Extend to multiplying by two-digit numbers.
Grid method
23 x 7 is approximately 20 x 10 = 200
72 x 38 is approximately 70 x 40 = 2,800
x
7
20
140
3
21
= 161
x
30
8
70 2
2100 60
560 16
= 2160
= 576
2736
Stage 6
x = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers
Expanded grid method
Extend to decimals with up to two decimal places.
4.68 x 6 is approximately 5 x 6 = 30
X
4
0.6
0.08
6
24
3.6
0.48
= 24
3.6
0.48
28.08
1
372 x 24 is approximately 400 x 20 = 8000
x
20
4
300
70 2
6000 1400 40 = 7440
1200 280 8 = 1488
8928
Expanded vertical written method
Extend to written method to cut down number of steps
used to solve a calculation.
38
x27
760
210
56
1026
1
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Division
=
Stage 1
PF - calculating
Solve practical problems that involve combining
groups of 2, 5 or 10, or sharing into equal groups
Stage 2
Stage 3
Division Objectives (excluding rapid recall)
Division Objjectives (excluding rapid recall)
PF - calculating
PF - calculating
Represent repeated addition and arrays as
Use practica
al and informal written methods to
multiplication, and sharing and repeated
multiply and
d divide two-digit numbers
subtraction (grouping) as division; use
(e.g. 13
practical and informal written methods and related
down, depen
nding on the context
3, 50
4); round remainders up or
vocabulary to support multiplication and division,
including calculations with remainders
Understand that division is the inverse of
multiplicatio
on and vice versa; use this to derive and
record relate
ed multiplication and division number
sentences
Find unit fra
actions of numbers and quantities (e.g.
,
PF-knowing & using number facts
Count on or back in ones, twos, fives and tens and
use this knowledge to derive the multiples of 2, 5
and 10 to the tenth multiple
,
and
of 12 litres)
PF-knowing & using number facts
PF-knowing & using number facts
Derive and recall multiplication facts for the 2, 5
Derive and recall
r
multiplication facts for the 2, 3, 4,
and 10 times-tables and the related division facts;
5, 6 and 10 times-tables and the corresponding
recognise multiples of 2, 5 and 10
division factts; recognise multiples of 2, 5 or 10 up
to 1000
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Division
Stage 1
Stage 2
Stage 3
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
÷ = signs and missing numbers
6÷2=
=6÷2
6÷ =3
3=6 ÷
÷2=3
3= ÷2
÷∇=3
3= ÷∇
Sharing – 6 sweets are shared between 2 people. How
many do they have each?
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?
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?
÷ = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers.
6 ÷ 2 can be modelled as:
3 groups
0
1
2
3
4
5
6
In the context of money count forwards and backwards
using 2p, 5p and 10p coins
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?
Practical grouping e.g. in PE
12 children get into teams of 4 to play a game. How many
teams are there?
Understand division as sharing and grouping
18 ÷ 3 can be modelled as:
Sharing – 18 shared between 3 (see Stage 1 diagram)
OR
Grouping - How many 3’s make 18?
0
3
6
9
12
15
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.
0
3
6
9
12
15 16
Expanded written method
Partition the dividend in to multiples of the divisor.
54 ÷ 3 = 18
30 + 24
÷3
10 + 8 =18
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Division
Division Objectives
Stage 4
PF – calculating
Develop and use written methods to record,
support and explain multiplication and division of
two-digit numbers by a one-digit number,
including division with remainders (e.g. 15
9, 98
Stage 5
Stage 6
Division Objectives
Division Objectives
PF – calculating
PF – calculating
Refine and use efficient written methods to multiply
Use efficien
nt written methods to add and subtract
and divide
integers and decimals, to multiply and divide
HTU
U, TU
and HTU
TU, U.t
U
U
integers an
nd decimals by a one-digit integer,
and to multtiply two-digit and three-digit integers
by a two-diigit integer
6)
PF-Knowing and using number
PF-Knowing and using number
facts
facts
Derive and recall multiplication facts up to 10 x 10,
Recall quickly multiplication facts up to 10 x 10 and
the corresponding division facts and multiples of
use them to multiply pairs of multiples of 10 and
numbers to 10 up to the tenth multiple
100; derive quickly corresponding division facts
REMEMBER: Children should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
Division
Stage 4
Stage 5
Stage 6
÷ = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers.
÷ = signs and missing numbers
Continue using a range of equations as in Stage 2 but with
appropriate numbers.
÷ = signs and missing numbers
Continue using a range of equations as in Stage 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 and grouping
Continue to understand division as both sharing and
grouping (repeated subtraction).
Sharing and grouping
Continue to understand division as both sharing and
grouping (repeated subtraction).
+6
0
+6
6
+6
12
+6
18
+6
24
Remainders
30
sharing – sharing among 6, the number given to each
person
Quotients expressed as fractions or decimal fractions
61 ÷ 4 = 15 ¼ or 15.25
+20
+40
Remainders
Quotients expressed as fractions or decimal fractions
676 ÷ 8 = 84.5
+32
+640
+1
80 groups
Remainders
41 ÷ 4 = 10 r1
10 groups
4 groups
5 groups
0
0
+40
+4
40
60
640
672
61
+1
Towards a compact written method
10 groups
Expanded written method
As with Stage 3 but include three-digit numbers as
the dividend and calculations with remainders.
41 = (10 x 4) + 1
Expanded written method
As with Stage 3 but include divisors of 7, 8 and 9.
977 ÷ 36
256 ÷ 7 = 36r4
210 + 46
÷7
Expanded written method with remainders
72 ÷ 5 = 16r2
30 + 6r4 = 36r4
50 + 22
÷5
REMEMBER:
10 + 4r2 Children
= 16r2 should have the opportunity to choose the appropriate strategy whether this is, mental strategies,
mental with jottings, calculator or written procedures.
676
Towards short division
From using the arrow method to find multiples of
the divisor to short division.
1)
Use the arrow method when dealing smaller
numbers.
54 ÷ 3 = 18
30 + 24
÷ 3
10 + 8 = 18
2)
Progress on to an expanded method of short
division when numbers become trickier to divide.
162 ÷ 6 = 27
10 + 10 + 7 = 27
6 60 + 60 + 42
3)
or
20 + 7 = 27
6 120 + 42
Learn the compact method for short division.
224 ÷ 7 = 32
3 2
7 2 2 4
We know that
7 x 30 = 210.
That leaves 14.
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