Addition – key objectives Step 1 Step 2 Step 3 Step 4 Step 5 1 + 2 = 3

Addition – key objectives
Step 1
Begin to relate addition to
combining two groups of objects
and subtraction to ‘taking away’
In practical activities and
discussion begin to use the
vocabulary involved in adding and
subtracting
Step 2
Step 3
Relate addition to counting on;
recognize that addition can be
done in any order; use practical
and informal written methods to
support the addition of a one-digit
number or a multiple of 10 to a one
digit or two-digit number.
Use vocabulary related to addition
and subtraction and symbols to
describe and record addition and
subtraction number sentences.
Use number songs and
stories, counting books
etc…
Practical recording eg.
•
•
+
Add or subtract mentally a onedigit number or a multiple of 10 to
or from any two-digit number; use
practical and informal written
methods to add and subtract twodigit numbers.
Understand that subtraction is the
inverse of addition and vice versa;
use this to derive and record
related addition and subtraction
number sentences.
Step 4
Step 5
Add or subtract mentally
combinations of one-digit and twodigit numbers.
Add or subtract mentally pairs of
two-digit whole numbers (eg. 47 +
58, 91-35).
Develop and use written methods
to record, support or explain
addition and subtraction of twodigit and three-digit numbers.
Refine and use efficient written
methods to add and subtract twodigit and three-digit whole
numbers and £.p
Use a calculator to carry out onestep and two-step calculations
involving all four operations;
recognize negative numbers in the
display, correct mistaken entries
and interpret the display
correctly in the context of
money.
Use the symbols +, -, x, ÷ and = to
record and interpret number
sentences involving all four
operations; calculate the value of
an unknown in a number sentence
(eg. ÷2=6, 30- =24)
ENL (bead strings)
•
•
“Hit the tens” method
Counting in ones on bead string
/frames
•
Number bonds to 10
(Hearts in love etc)
•
ENL jumping to the
next ten.
•
+3
+10
+6
-1
37
66
=
37
40
50
13
+2
1+2=3
8
60
37 + 29 = 66
66
37 + 29 = 66
+3
10
13
8 + 5 = 13
•
•
•
•
Dice games
Number tracks
Introduce bead strings
Counting in ones from
numbers to ten.
•
“Tens jumping”
(preferred method)
•
66
+10
+10
37
47
+3
57
37 + 29 = 66
+6
60
66
•
66
+30
66
+10
“Over jumping”
Partitioning
37 = 30 + 7
29 = 20 + 9
30+ 20 = 50
7 + 9 = 16
So
37+29 = 66
67
“Expanded method”
37
+29
50 (30+20)
16 (7+9)
66
Addition – key objectives
Step 6
Step 7
Use efficient written methods to
add and subtract whole numbers
and decimals with up to two places.
Use efficient written methods to
add and subtract integers and
decimals.
Extend mental methods for whole
number calculations.
Use a calculator to solve problems
involving multi-step calculations.
•
“Compact method”
37
+29
66
1
Calculate mentally with integers
and decimals: U.t ± U.t, TUxU,
TU÷U, U.txU, U.t÷U
•
“Compact method”
126.4
+115.3
241.7
1
Step 8
Extend carrying method to
numbers with at least four digits,
including decimals in a range of
contexts. Eg. Money and
measurement.
•
Adding decimals with a
mixed number of decimal
places.
3.59 + 0.7 + 5
3.59
0.70
+5.00
9.29
1
Step 9
Step 10
Extend understanding of addition
to add fractions with the same
denominator and fractions with
denominators of the same number.
•
Add fractions with equal
denominators.
2+2=4
5 5 5
4+5=9=
6 6 6
Extend understanding of addition
and equivalent fractions to add
fractions with different
denominators and mixed numbers.
•
Add fractions with
unequal denominators.
1+1=4+1=5
2 8 8 8 8
1
3
6
2 + 3 = 8 + 9 = 17 =
3 4 12 12 12
1125
Subtraction – key objectives
Step 1
Step 2
Begin to relate addition to
combining two groups of objects
and subtraction to ‘taking away’
In practical activities and
discussion begin to use the
vocabulary involved in adding and
subtracting
Step 3
Understand subtraction as “take
away” and find a “difference” by
counting up; use practical and
informal written methods to
support the subtraction of a onedigit number from a one digit or
two-digit number and a multiple of
ten from a two-digit number.
Use vocabulary related to addition
and subtraction and symbols to
describe and record addition and
subtraction number sentences.
•
•
•
•
•
Use number songs and
stories, counting books
etc…
Dice games
Number tracks
Introduce bead strings.
Practical recording eg.
•
•
•
•
•
•
-
Number bonds to 10
(Hearts in love etc)
Introduce vocabulary of
subtraction.
Bead strings to assist
mental calculation.
Use +/-/= to record
calculation.
Use number line for
counting back.
Use addition/ counting
on to solve subtraction,
find the difference.
Understand that subtraction is the
inverse of addition and vice versa; use
this to derive and record related
addition and subtraction number
sentences.
•
•
•
ENL (bead strings)
Counting back
Counting on / find the
difference.
3
=
+1
2
3-1=2
Step 4
Add or subtract mentally a one-digit
number or a multiple of 10 to or from
any two-digit number; use practical and
informal written methods to add and
subtract two-digit numbers.
+1
3
5-2=3
+10
+2
5
Develop and use written methods to
record, support or explain addition
and subtraction of two-digit and
three-digit numbers.
•
•
Consolidate ENL methods
used in Y2.
Use number complements
to make larger jumps.
51
15
+1
4
Step 5
Add or subtract mentally
combinations of one-digit and twodigit numbers.
18
20
+1
+3
30
33 - 18 = 15
+50
33
39
40
90 - 39 = 51
90
Add or subtract
mentally pairs of twodigit whole numbers (eg.
47 + 58, 91-35).
Refine and use efficient
written methods to add
and subtract two-digit
and three-digit whole
numbers and £.p
Use a calculator to
carry out one-step and
two-step calculations
involving all four
operations; recognize
negative numbers in the
display, correct
mistaken entries and
interpret the display
correctly in the context
of money.
•
Convert ENL
into column
methods of
recording.
•
Many pupils
may continue
to use
number line
if more
comfortable
withy this.
130
- 79
21 (100)
30 (130)
51
Subtraction – key objectives
Step 6
Step 7
Use efficient written methods to
add and subtract whole numbers
and decimals with up to two places.
Extend mental methods for whole
number calculations for example to
subtract one near multiple of a
thousand from another (eg. 60704097)
•
Extend column
subtraction with larger
numbers and decimals.
675
-136
4 (140)
60 (200)
400 (600)
75 (675)
539
Step 9
Step 8
Use efficient written methods to
add and subtract integers and
decimals, to multiply and divide
integers and decimals by a onedigit integer, and to multiply two
digit and three digit integers by a
two digit integer.
Use understanding of subtraction
to subtract fractions with the
same denominator.
Step 10
Apply understanding of
subtraction and equivalent
fractions to subtract fractions
with different denominators and
mixed numbers.
Use a calculator to solve problems
involving multi-step calculations.
•
600
Introduce expanded
decomposition.
140
700 40 9
- 200 50 2
400 90 7
Calculate mentally with integers
and decimals: U.t ± U.t, TUxU,
TU÷U, U.txU, U.t÷U
•
Move towards Compact
decomposition including
decimals.
2
•
Subtract fractions.
•
4–2=2
5 5 5
1–1=4–1 =3
2 8 8 8 8
1
36.57
- 17.46
19.11
Subtract fractions with
unequal denominators.
2
1 – 3 = 10 – 3
2 4 4 4
=7=
4
1 43
Multiplication – key objectives
Step 1
Step 2
Count repeated groups of the
same size.
Solve practical problems that
involve combining groups of 2, 5
or 10, or sharing into equal
groups.
Know doubles of all numbers to
ten.
Step 3
Step 4
Step 5
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.
Derive and recall multiplication facts for
the 2, 3, 4, 5, 6 and 10 times-tables and
the corresponding division facts;
recognize multiples of 2, 5 or 10 up to
1000. Multiply one-digit and two-digit
numbers by 10 or 100, and describe the
effect.
Know by heart all multiplication
facts for times table up to 10 x 10.
Use the symbols +, -, x, ÷ and = to
record and interpret number
sentences involving all four
operations; calculate the value of
an unknown in a number sentence
(eg ÷ 2 = 6, 30 - = 24)
•
Count objects in sets,
be able to say
whether sets are the
same, larger or
smaller.
•
•
•
Learn double to
twenty
Count on and back in
2s, 5s and 10s.
Using bead frames
look at the pattern
of five.
•
•
•
Multiplication as
repeated addition
4x3=4+4+4
Use number lines to
solve problems with
missing numbers eg.
Use practical and informal written
methods to multiply and divide two-digit
numbers (eg. 13 x 3, 50 ÷ 4); round
remainders up or down, depending on the
context.
Understand that division is the inverse
of multiplication and vice versa; use this
to derive and record related
multiplication and division number
sentences.
•
Understand multiplication is
the inverse of division.
•
Partitioning.
0
2x2
Develop and use written methods
to record, support and explain
multiplication and division of twodigit numbers by a one-digit
number, including division with
remainders (eg. 15 x 9, 98 ÷ 6)
•
Grid method of
multiplication, then
develop into more
formal recording.
24 x 6
x 20
x2=6
1x2
Multiply and divide numbers to
1000 by 10 and then 100 (wholenumber answers), understanding
the effect; relate to scaling up or
down.
4
6 120 24
3x2
144
6
•
•
Learn multiplication
facts for 2s, 5s and 10s
by heart.
Introduce formal
recording and x sign.
13 x 3 =
10 x 3 = 30
3x3= 9
39
•
Work with place value
charts x10 and x100
H T U
6 2 x 10
6 2 0
Multiplication – key objectives
Step 6
Step 7
Step 8
Extend mental methods for wholenumber calculations, for example
to multiply a two-digit by a onedigit number (eg. 12 x 9) to
multiply by 25 (eg. 16 x 25).
Use knowledge of place value and
multiplication facts to 10 x 10 to
derive related multiplication and
division facts involving decimals (eg.
0.8 x 7, 4.8 ÷ 6)
Use understanding of place value
to multiply and divide whole
numbers and decimals by 10, 100
or 1000.
Use knowledge of multiplication
facts to derive quickly squares of
numbers to 12 x 12 and the
corresponding squares of multiples
of 10.
Refine and use efficient written
methods to multiply and divide
HTU xU, TU x TU, U.t x U
•
Extend grid method
34 x 56
x 30 4
50 1500 200 1700
6 180 24
204
Use efficient written methods to
multiply integers and decimals by a
one-digit integer, and to multiply
two-digit and three-digit integers
by a two-digit number.
Further extend grid
•
method.
•
Develop into expanded
column method.
56
x 34
1500 (30x50)
180 (30 x 6)
200 (50 x 4)
24 (6 x 4)
1904
Extend the compact method to
ThHTUxU, ThHTUxTU and
decimals up to 3 decimal places.
•
Short multiplication
342
X
7
2394
•
Long multiplication
124
X 26
2480
744
3224
•
3
30 6000 1200
90
7290
6 1200 240
18
1458
Apply understanding in the use of
formal written methods for short
and long multiplication and to
multiply pairs of proper fractions.
Short multiplication
4.38
X
7
30.66
21
x 200 40
Step 10
Apply their knowledge and
understanding of multiplication
and use materials and diagrams to
multiply proper fractions and
mixed numbers by whole numbers.
243 x 36
8748
1904
Step 9
2 5
•
Long multiplication
16.741
X 35
502.23
83.705
585.935
•
Multiply proper
fractions by whole
numbers.
1 1
•
Compact column method
56
x 34
1680 (56x30)
224 (56 x 4)
1904
•
Short multiplication
2741
X
6
16446
4 2
•
Long multiplication
3124
X 26
62480
18744
81224
111
2 x
3
3= 2
2 12 x 3 = 7 21
•
Multiply pairs of proper
fractions.
2 x 4= 8
3 5 15
Division – key objectives
Step 1
Share objects into equal groups
and count how many in each group.
Step 2
Step 3
Step 4
Solve practical problems that
involve combining groups of 2, 5 or
10 or sharing into equal groups.
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.
Use practical and informal written
methods to multiply and divide
two-digit numbers (eg 13x3, 50÷4);
round remainders up or down,
depending on the context.
Multiply and divide numbers to
1000 by 10 and then 100 (wholenumber answers), understanding
the effect; relate to scaling up or
down.
Understand that division is the
inverse of multiplication and vice
versa; use this to derive and
record related multiplication and
division number sentences.
Develop and use written methods
to record, support and explain
multiplication and division of twodigit numbers by a one-digit
number, including division with
remainders (eg 15x9, 98÷6)
Use the symbols +, -, x, ÷ and = to
record and interpret number
sentences involving all four
operations.
Step 5
Find unit fractions of numbers and
quantities (eg. ½, ¼ and ⅛ of 16
litres)
Find fractions of numbers,
quantities or shapes (eg ⅛ of 24
plums, ⅜ of a 6 by 4 rectangle)
Use a calculator to carry out onestep and two-step calculations
involving all four operations;
correct mistaken entries and
interpret the display correctly.
•
•
Link to doubling and
halving
Know halves of even
numbers to 20
•
•
Introduce division
symbol ÷
Division as sharing …
14÷3 = 4 r 2
•
Inverse operation
4x3=12
•
So …
•
Use knowledge of times
tables facts to divide on
ENL
12÷3=4
Counting on/back on ENL
12
10x6
-3
Sharing
•
Division as grouping (this
is more useful as an
introduction to ENL)
14÷3 = 4 r 2
0
-3
-3
0
-3
12
2x6
60
72
Division – key objectives
Step 6
Step 7
Use understanding of place
value to multiply and divide
whole numbers and decimals by
10, 100 or 1000.
Refine and use efficient
written methods to multiply
and divide HTU x U, TUxTU,
U.txU and HTU ÷U
Find fractions using division (eg
1/100 of 5kg) and percentages
of numbers and quantities (eg
10%, 5% and 15% of £80)
Use a calculator to solve
problems, including those
involving decimals or fractions
(eg find ¾ of 150g); interpret
the display correctly.
•
Lead into more formal
recording of “Target method”
56 56÷4=14
14x4 56
10x4 40
4x4 16
2x4 8
1x4 4
0
Continue to develop
“Target method”,
refine to make steps
more efficient.
•
182
182÷7=26
26x7 182
25x7
20x7
10x7
5x7
175
140
70
35
1x7 7
0
Step 8
Step 9
Step 10
Calculate mentally with integers
and decimals: U.t±U.t, TUxU,
TU÷U, U.txU, U.t ÷U
Use efficient written methods to
add and subtract integers and
decimals, to multiply and divide
integers and decimals by a onedigit integer, and to multiply twodigit and three-digit integers by a
two-digit integer.
Relate fractions to multiplication
and division (eg 6÷2= ½of6 = 6 x
½); express a quotient as a
fraction or decimal (eg 67÷4 =
16.75 or 16¾); find fractions and
percentages of whole-number
quantities (eg ⅝ of 96, 65% of
£260)
Use a calculator to solve problems
involving multi-step calculations.
•
Lead into more formal
recording.
12
6 72
60 (10 x 6)
12 (2 x6)
•
Continue to develop
compact methods.
12
6 72
•
Include decimals
2.14
7 14.98
Division – key objectives
Step 11
Children will use their understanding of
the relationship between multiplication and
division to apply short division for whole
numbers up to ThHTU÷U and interpret
remainders in context.
•
Continue to develop compact
methods.
Step 12
Children should extend
understanding to convert
remainders to fractional OR
decimals as requested.
•
Include decimals
432
449
2 1
5
8 3456
11
12 5391
449 r 3
5
3
12
11
12 5391
449.25
5
11
3
6
12 5391.00
Step 13
Step 14
Step 15
Children use their understanding
of division to use a formal written
method to divide numbers up to 4
digits by 2 digits. Any remainders
should be shown as fractions when
requested.
Children should also apply their
knowledge and understanding of
division and use materials and
diagrams to divide proper
fractions by whole numbers .
•
Include decimals
28 r 12
15 432
300 (20x15)
132
120 (8x15)
12
•
Include decimals
28
15 432
300 (20x15)
132
120 (8x15)
12
4
5
12 = 4
15 5
•
Include decimals
28.8
15432.0
30
132
120
12.0
12.0
0
Calculation Strategy
Lantern Community Primary School
2014
Foundation Stage
Key Stage 1
Key Stage 2
Foundation to Year 6
Pencil and paper procedures
This policy contains the key pencil and paper procedures that will be taught at the Lantern. It has
been written to ensure consistency and progression throughout the school.
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 Numeracy Framework. The mental methods
in the Teaching children to calculate mentally document will be taught systematically from Reception
onwards and pupils will be given regular opportunities to practise 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 the children to think about the mental
strategies that underpin them 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 the time at school children will be encouraged to see mathematics both as 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 mental strategy;
• Use of pencil and paper procedures;
• 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 pencil and paper procedures. It is
important that children DO NOT abandon jottings and mental methods once pencil and paper
procedures are introduced. Therefore children will always be encouraged to look at a
calculation/problem and then decide what 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 that is appropriate for a given task. They will
do this by always asking themselves:
‘Can I do this in my head?’
‘Can I do this in my head using drawings or jottings?’
‘Do I need to use a pencil and paper procedure?’
‘Do I need a calculator?’
The following pages aim to provide a quick reference guide for teachers illustrating how these
procedures are developed stage by stage. It is important to realise that pupils must not abandon
stages until they can demonstrate that they fully understand it and can employ it successfully in a
range of contexts. These steps are not a direct substitute for year by year progression. Young
mathematicians can have their confidence dashed very easily if they are forced into rigorous recipe
methods that they don’t understand, are unclear and have no connection to prior learning. As a
result the use of the empty number line can be seen as a firm basis for all four operations in any
year as appropriate.
Progression in calculation
Key skills
•
•
•
•
•
•
•
•
Develop counting and the language of counting
Develop understanding of the process of counting and the number system
Secure conceptual understanding of place value
Use knowledge of known facts
Develop confidence in mental calculation skills
Understand operations and the relationship between them
Foster mathematical thinking
Develop skills in recording including jottings
Counting development
• Enactive – actions, practical, real life context
• Iconic – representation, models, images, pictures
•
Symbolic – recording, numbers, letters
Stages
•
•
•
•
•
•
Counting all
Counting on
Counting on from the first number
Counting on from the larger number
Using a known fact
Using knowledge of place value
Preparing for place value
From Iconic to symbolic: Use concrete objects, practical experiences and pictures to
develop conceptual understanding of place value. Progression using models, images and
pictures will secure symbolic understanding and secure knowledge of place value.
Towards Calculation
Confidence in mental calculation skills
Conceptual understanding of place value is secure
Practise and use the skill of estimation
Understand the process being used
Compare and evaluate methods
Develop a broad range of experiences in recording calculations – Can I do this in my
head? Do I need a jotting? Do I need a written method?
• Foster mathematical thinking and discussion
• Make the maths real – link mathematics across the curriculum and to the outdoors
•
•
•
•
•
•