Luckwell Primary School
Calculation Strategy
Models and Images and Known Facts
Concrete-symbols-abstract
Gemma Fricker
2014
Array- a regular arrangement of objects
4 x 3 or 3 x 4
Base 10
Base 10- cubes used to give a visual representation of 1, 10, 100 and 1000
Bridging (through 10/100)-to use knowledge of number bonds to count on or back to the nearest/best 10 as an
efficient strategy to get to the final number e.g. 13 + 8 as 13 +7 = 20 + 1 = 21
Cardinality-the size (value) of a number compared to another number
Commutativity-that addition or multiplication can be done in any order e.g. 2X3 is the same as 3X2
Compensation- to use known number facts to calculate a number that is near to the one known e.g. 27 + 9 as 27 + 10
then adjust (compensate) by subtracting 1
Cuisenaire-different coloured rods that can be used to represent any number or value
Cuisenaire
Denominator-the bottom number in a fraction representing the number of parts that make up the whole
Distributivity-a law used in algebra e.g. 3(2a+4c) = 6a +12 can be understood as 3 lots of 2 times the first number
added to 4 times another number is the same as 6 lots of the first number add 12
Factor-a whole number that multiplies with another number to make a third number e.g. 2 and 5 are factors of 10
Integers-a complete number, not a fraction or decimal e.g. 3, -3, 103
Inverse- opposite/reverse operations e.g. 4 + 6= 10 10 – 4= 6
Numerator-the top number in a fraction representing the number of equal parts out of the whole
Numicon- shapes that give a visual representation of numbers from 1-10 and can be
combined to make larger numbers
Numicon
Ordinal number-1st, 2nd, 3rd etc
Proportionality-the relationship between two variables that remains constant e.g. in enlarging a piece of A4 paper the
long side will always be 2 times the length of the short side
For more definitions visit:
http://www.amathsdictionaryforkids.com/dictionary.html
Guidance
This strategy has calculation methods for all 4 operations and is designed to be used in a linear progression. The year
groups attached are guidelines in line with the Nation Curriculum 2012, however children should use the strategy
appropriate to their development as assessed by the class teacher.
Secure knowledge of number facts and in particular place value are key to children’s success in mastering written
methods and should be a priority
By the end of year 6, children will have a range of calculation methods, mental and written. Selection will depend
upon the numbers involved.
Children should not move onto the next stage if they are not ready or they are not confident and show a lack of
understanding of the method.
Children should be encouraged to approximate their answers before calculating.
Children should be encouraged to consider if a mental calculation would be appropriate before using written
methods.
M/O Facts
R
e
c
Addition
Recognise numbers Children are encouraged to develop a
0-5, then 0-10, to 20 mental picture of the number system
in their heads to use for calculation.
Count with and
order numbers to 20 Children should count with and be
able to order nos to 20
Find 1 more and 1
less than numbers
to 5, then 10, to 20
Some number
bonds
Record marks
they can interpret
and explain
Recognise odd and
even numbers
using apparatus
Doubles and
halves in context
and using
apparatus
The cardinality of numbers must be
made explicit.
e.g.
The ‘eightness’ of 8
Subtraction
Children are encouraged to
develop a mental picture of the
number system in their heads to
use for calculation.
They develop ways of recording
calculations using pictures,
objects etc.
Use Numicon covers to ‘hide’
parts e.g.
Using quantities and objects, they
add two single-digit numbers and
count on to find the answer.
They develop ways of recording
calculations using images, pictures,
objects etc.
Multiplication
6-2=4
Say the number that is 1 less
than a number to 20
In practical activities begin to
use the vocabulary involved in
subtraction
+
=
Say the number that is 1 more than a
number to 20
In practical activities begin to use
the
vocabulary involved in addition
Teachers model the use of the
number line
Bead strings or bead bars can be
used to illustrate subtraction.
6-2=4
.
Children will experience equal
groups of objects.
In songs and rhymes, they will
count in 2s and 10s and begin to
count in 5s.
They solve problems, including
doubling
Overlaying Numicon
They will work on practical
problem solving activities
involving equal sets or groups.
Division
Children will understand equal
groups and share items out in play
and problem solving.
They solve problems, including
halving and sharing.
Overlaying Numicon
Y
1
Count to and across
Use numbers to 20
100 forwards and
Add and subtract 1-digit numbers to
back, from 1, 0 or
20, inc 0
any number, inc in
ones, twos, fives and Record their work using pictures and
tens
images
Read and write nos
to 20 in numerals
and words
Identify and
represent numbers
using objects and
pictorial
representations inc
the number line and
use language equal
to, more than, less
than, fewer, most
and least
Represent and use
number bonds within
20 and related
subtraction facts
4+6
Doubling and halving
Recognise odd and
even numbers
Use numbers to 20
Record their work using pictures and
images
6-2=4
Solve one-step problems involving
multiplication and division using
concrete objects, pictorial
representations and arrays with
the support of the teacher
Use Numicon covers to ‘hide’ parts
+
4x3=12
9÷3=3
=
Begin to use symbols = - + < and > to
record L1 calculations
Begin to use symbols = - + < and > to
record L1 calculations
Partition numbers in different ways
e.g. 7=3+4, 7=2+5, 7-3=4
Solve one-step problems involving
addition and subtraction using
concrete objects, pictorial
representations , and missing
number problems
e.g 7 =
- 9
Memorise and reason with number
bonds in different forms
e.g. 3+7=10, 10=3+7, 10-3=7
Bead strings or bead bars can be
used to illustrate subtraction
including modelling how to bridge
through ten by counting back 3 then
counting back 2.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Children begin to use numbered
number lines to support their own
calculations counting on in ones
Bead strings or bead bars can be
used to illustrate addition inc
modelling how to bridge through 10
8+5 (8+2+3)=10
Solve one-step problems involving
multiplication and division using
concrete objects, pictorial
representations and arrays with
the support of the teacher
13-5=8
Children use numbered number lines
to support their own calculations
counting back in ones.
The number line should also be used
to show that 6 - 3 means the
‘difference between 6 and 3’ or ‘the
difference between 3 and 6’ and how
many jumps they are apart by
counting on.
Using measures as a context
supports this notion e.g. difference
between lengths
They make connections between
arrays, number patterns and
counting in 2s, 5s and 10s
Through grouping and sharing small
quantities pupils begin to
understand multiplication...doubling
numbers and quantities
This should be in the context of
measures as well as numbers
e.g. 4 cups of rice is equal to 1 jug
of rice, 1 apple is equal in mass to 3
satsumas
ie. 3x Satsuma=apple
12÷3=4
Recognise, find and name a half as
one of two equal parts of an object
or a shape or quantity
Recognise, find and name a quarter as
one of four equal parts of an object
or a shape or quantity
Through grouping and sharing small
quantities pupils begin to understand
division...finding simple fractions of
objects and quantities
This should be in the context of
measures as well as numbers
e.g. 4 cups of rice is equal to 1 jug of
rice, 1 apple is equal in mass to 3
satsumas
Mental/Oral
Facts
Y
2
Count in steps of
2,3, 5, 0 and 10
from any number
forwards and back
Recall and use
addition facts to
20; Using pictorial
representations,
inc numbers,
quantities and
measures to
derive facts e.g.
subtraction facts
to 20 20/100]
Addition
Subtraction
Add and subtract numbers using
concrete objects, pictorial
representations and mentally, inc. a
2-digit number and ones; a 2-digit
number and tens; 2 2-digit numbers
e.g.
+
=
Children will begin to use empty
number lines to support calculations.
=
0r
Y
2
Count, read and
write nos to 100 in
numerals and
words
Use PV and
number facts to
solve problems
+
Division
Children will develop their
understanding of multiplication and
use jottings to support calculation:
Repeated addition
3 times 5 is 5 + 5 + 5 = 15 or 3
lots of 5 or 5 x3
Repeated addition can be shown
easily on a number line:
Children will develop their
understanding of division and use
jottings to support calculation
Sharing equally
6 sweets shared between 2 people,
how many do they each get?
Grouping or repeated subtraction
There are 6 sweets, how many people
can have 2 sweets each?
and on a bead bar:
Counting back:
First counting back in tens and ones.
Or with Numicon:
Compare and
order numbers
from 0-100 use < >
and = signs
Recognise the
place value of
each digit in a 2digit number
Add and subtract numbers using
concrete objects, pictorial
representations and mentally, inc. a
2-digit number and ones; a 2-digit
number and tens; 2 2-digit numbers
Multiplication
=
Using known facts:
Then helping children to become more
efficient by subtracting the units in
one jump (by using the known fact 7 –
3 = 4).
Subtracting the tens in one jump and
the units in one jump.
3X3=9
Commutativity:
Children should know that 3 x 5 has
the same answer as 5 x 3.
Numicon shapes illustrate this well
It can also be shown on the number
line.
Repeated subtraction using a
number line or bead bar
12 ÷ 3 = 4
Or Numicon:
9÷3=3
Children will begin to use ‘empty
number lines’ themselves starting
with the larger number and counting
on.
PTO
First counting on in tens and ones.
Bridging through ten can help
children become more efficient.
Arrays
Children should be able to model a
Using symbols to stand for unknown
numbers to complete equations using
inverse operations
3X5=15 so 15÷ =3 must be 5
Use x and ÷ signs to write
multiplication and division statements
Recall and use
multiplication and
division facts for
the 2, 5 and 10
times tables
Recognise odd and
even numbers
Then helping children to become
more efficient by adding the units in
one jump (by using the known fact 4 +
3 = 7).
Y
2
Followed by adding the tens in one
jump and the units in one jump.
Bridging through ten can help
children become more efficient.
Partitioning numbers in different
ways
e.g. 17=13+4, 17=2+15, 17-3=14
Partitioning to add using base 10
materials as support
Adding ‘tens’ and ‘ones’
e.g. 34+23
30 + 4
20 + 3
50 + 7
Commutativity
Addition can be done in any order
2+3=5 and 3+2=5
Recording in different ways 5=3+2 0r
1+4=2+3; 3+4>5
multiplication calculation using an
array. This knowledge will support
with the development of the grid
method.
Counting on:
The number line should still show 0 so
children can cross out the section
from 0 to the smallest number. They
then associate this method with
‘taking away’.
Children can be given a ‘rule’ for when
to count on to ‘find the difference.’
Measures are a useful context e.g.
long jumps, lengths of rope
Partitioning and decomposition
Using base 10 materials
e.g. 89-57=32
80 + 9
- 50 + 7
30 + 2
Children would not be expected to
record this as column subtraction in
Year 2
Recognise and use the inverse
relationship between addition and
subtraction and use this to check
calculations and solve missing number
problems
5+7=12 so 12-
=5 must be 7
Using symbols to stand for unknown
numbers to complete equations using
inverse operations
Use x and ÷ signs to write
multiplication and division statements
4 x5=20 and 20÷4=5
Solve problems involving
multiplication and division using,
material,arrays, repeated addition,
mental methods and multiplication
and division facts, including
problems in context
4 x5=20 and 20÷4=5
Using symbols to stand for
unknown numbers to complete
equations using inverse operations
 ÷ 2 = 4 20 ÷  = 4
÷=4
Solve problems involving
multiplication and division using,
material,arrays, repeated addition,
mental methods and multiplication
and division facts, including
problems in context
M/O facts
Read, write
Y3 (numbers and
words) order and
compare and
order numbers up
to 1000
Using a variety of
representations,
including
measures, count
in ones, tens and
hundreds, so that
they become
fluent in the
order and place
value of numbers
to 1000
Count on from
zero in multiples
of 4, 8 50 and
100
Find 10 or 100
more or less than
a given number
Addition
Children will continue to use empty
number lines with increasingly large
numbers, including compensation
where appropriate.
Count on from the largest number
irrespective of the order of the
calculation.
Compensation
Adding ‘tens’ then ‘ones’.
Children will begin to use informal
pencil and paper methods (jottings)
to support, record and explain partial
mental methods building on existing
mental strategies (until secure in
strategy). Base 10 should be used
alongside written jottings.
Adding the least significant digits
first
Subtraction
Children will continue to use empty
number lines with increasingly large
numbers.
Partitioning and decomposition
 Partitioning – demonstrated using
arrow cards
 Decomposition - Base 10 materials
NOTE When solving the calculation
89 – 57, children should know that 57
does NOT EXIST AS AN AMOUNT
it is what you are subtracting from
the other number. Therefore, when
using base 10 materials, children
would need to count out only the 89.
Begin to exchange.(‘steal’ not
‘borrow)’
Multiplication
Division
Using multiplication tables that they
know:
Children will continue to use:
Repeated addition
4 times 6 is 6 + 6 + 6 + 6 = 24
or 4 lots of 6 or 6 x 4
Children should use number lines or
bead bars to support their
understanding.
(Teaching bead bars are available.)
Cuisenaire rods can be used to
support multiplicative relation ships
ie. If this (rod) is 3 then what is this
(2x or 3x larger rod)?
Commutativity
Children should know that 3 x 5 has
the same answer as 5 x 3.
Numicon pieces can illustrate this
clearly.
This can also be shown on the number
line.
Arrays
Children should be able to model a
multiplication calculation using an
array. This knowledge will support
with the development of the grid
method.
Then add numbers up to 3-digits
Recognise the
using formal written methods of
value of each
column addition.
digit in a 3-digit
number
Estimate answers to calculations
e.g. use inverse operations to check
answers
Solve problems including missing
number problems using number facts,
place value and more complex
addition and subtraction
Finding the difference
Where the numbers are involved in
the calculation are close together or
Scaling
near to multiples of 10, 100 etc
e.g. Find a ribbon that is 4 times
counting on using a number line should
as long as the blue ribbon
be used.
PTO
You could also use capacity or
Cuisenaire rods to develop this
Using multiplication tables that they
know:
Ensure that the emphasis in Y3 is on
grouping rather than sharing.
Children will continue to use:
Repeated subtraction using a
number line
Cuisenaire rods can be used to
support multiplicative relation ships
ie. If this (rod) is 8 then what is
this (2x or 3x smaller rod)?
Children should also move onto
calculations involving remainders.
Using symbols to stand for
unknown numbers to complete
equations using inverse operations
26 ÷ 2 =  24 ÷  = 12  ÷ 10 = 8
Scaling
Use bar method
Scaling should be used to solve
problems in context e.g. Find …4
times shorter than this…
Or James has saved £8. Sarah has
saved twice as much. How much
have they saved altogether?
a
Fractions
Recognise, find and write fractions
of discrete sets of objects: unit
fractions and non-unit fractions
with small denominators. This should
go beyond 0, 1
Y3 Add and subtract
mentally
including a 3digit number and
ones, a 3-digit
number and
tens, , a 3-digit
number and
hundreds
Multiplication
facts for (2,5,10)
3,4,6 and 8 x
tables up to
12X…
Division facts for
times tables
2,5,10,3,4,6
e.g.30 ÷ 6= 5;
6 = 30÷5
Count up and
down in
tenths…by
dividing an
object and in
dividing 1-digit
numbers or
quantities by 10
Working within the context of
measures supports this notion.
Can support this using Cuisenaire (bar
method)
Scaling should be used to solve
problems in context e.g. Find …4
times as long as this…
Using symbols to stand for unknown
numbers to complete equations using
inverse operations
 x 5 = 20 3 x  = 18  x  = 32
Partitioning
Jottings should be used to support
this. Some children will require
concrete apparatus at this point.
38 x 5 = (30 x 5) + (8 x 5)
= 150 + 40
= 190
Counters numbered in 1s, 10s, or 100s
could be used
Use measures as context e.g. ½ of 2
metres; ¼ of 2 metres
Recognise and use fractions as
numbers: unit fractions and non-unit
fractions with small denominators
E.g Fractions of shapes
Or Bar method e.g. NNS ITP
fractions
Recognise and show with diagrams
equivalent fractions with small
denominators
Children should then, when ready,
move to reliable written methods
Grid method:
Add and subtract fractions with the
same denominators within 1 whole
e.g. 5/7 + 1/7 = 6/7
The grid method is important as it
leads into the partitioning model
(below) for mental multiplication and
division of 2 digit by 1 digit numbers
2-digit by 1-digit numbers
Compare and order unit fractions
and fractions with the same
denominator
e.g. Using measures 1 ½ litres, ¼
litre, 1 1/3 litre, ¾
Recall
multiplication and
Y4 division facts for
multiplication
tables to 12x12
Count on from
zero in multiples
of 6, 7, 9, 25 and
1000
Count back
through zero to
negative numbers
Add numbers with up to 4 digits
using column addition where
appropriate *
Subtract numbers with up to 4
digits using column subtraction
where appropriate*
*Children should be taught to
recognise where metal strategies
would be more efficient
*Children should be taught to
recognise where metal strategies
would be more efficient
Carry below the line.
Partitioning and decomposition
Adjusting from ‘tens’ to ‘ones’
and ‘hundreds’ to tens’
Base 10 or place value counters
should be used alongside written
algorithm to support children
Base 10 or place value counters
should be used alongside written
algorithms to support children
Use place value and known and
derived facts to multiply and
divide mentally inc. multiplying by
1 and 1 dividing by 1 and
multiplying together 3 numbers
e.g. 200 × 3 = 600 into 600 ÷ 3 =
200, to become fluent.
Children will continue to use
arrays where appropriate leading
into the grid method of
multiplication.
Modelling the two side-by-side
can secure understanding.
Use place value and known and
derived facts to multiply and
divide mentally inc. multiplying
by 1 and 1 dividing by 1
e.g. 600 ÷ 3 = 200 can be
derived from 2X3=6
Children will develop their use of
repeated subtraction to be able
to subtract multiples of the
divisor. Initially, these should
be multiples of 10s, 5s, 2s and 1s
– numbers with which the
children are more familiar.
Recognise the
value of each
digit in a 4-digit
number
Order and
compare numbers
beyond 1000
Identify,
represent and
estimate numbers
using different
representations
(inc measures)
Round any
number to the
nearest 10, 100
or 1000(inc
measures)
Children should continue to
practice both mental and written
strategies with increasingly large
numbers as well as mixed numbers
of digits to aid fluency
Children should know that the
decimal points should line up
under each other, particularly
when adding or subtracting mixed
amounts, e.g. £3.59 + 78p.
Children must be taught to
Decomposition
Children should be able to
subtract numbers with different
numbers of digits and know that
decimal points should line up
under each other.
Grid method
Multiply 2-digit and 3-digit
numbers by a 1-digit number using
formal written methods
TU x O (to HTO x TO)
(Short multiplication –
multiplication by a single digit)
23 x 8
Children will approximate first
23 x 8 is approximately 25 x 8 =
200
Then onto the vertical method:
Short division TU ÷ O
Showing working alongside.
Leading to subtraction of other
multiples.
(PTO)
Any remainders should be shown
Y4 Read Roman
numerals to 100
and know that
over time the
number system
changes to
include the
concept of zero
Count up or down
in hundredths,
recognise that
hundredths arise
when dividing and
object by 100
and dividing
tenths by ten
Count in simple
fractions
estimate and use inverse
operations to check a calculation
Children must be taught to
estimate and use inverse
operations to check a calculation
HTO x O
(Short multiplication –
multiplication by a single digit)
346 x 9
Children will approximate first
346 x 9 is approximately 350 x 10
= 3500
as integers, i.e. 14 remainder 2
or 14 r 2.
Children need to be able to
decide what to do after division
and round up or down
accordingly. They should make
sensible decisions about rounding
up or down after division.
Complete equations using inverse
operations
Complete equations using inverse
operations
Integer scaling problems
e.g. scaling up shapes, models,
linear measures.
Correspondence problems such as
n objects are connected to m
objects
Eg. Three cakes shared equally
between 10 children
Fractions
Recognise and show using
diagrams, families of equivalent
fractions
Use bar and linear model
Cuisenaire rods can support this
model
number diagrams to support as
well as going beyond 0-1 eg. 0-2
number lines
Add and subtract fractions with
the same denominator
Recognise and write decimal
equivalents of any number of
tenths or hundredths
Measures are a good context
here.
Recognise and write decimal
equivalents to ¼, ½ ¾
Use liquids to aid visualisation
Y4
Find the effect of dividing a 1 or
2-digit number by 10 and 100
Aid memoir e.g. ‘Zero the Super
Hero’.
Round decimals with 1 decimal
place to the nearest whole
number
Compare numbers with the same
number of decimal places to the
nearest two decimal place