Progression in Calculations Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. These lead on to more for

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Progression in Maths - calculation methods
Abbey Primary School December 2012
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Introduction

• Written methods of calculations are reliable
  and efficient procedures which can be used in
  many different contexts. However, they are of no
  use to someone who applies them inaccurately or
  who cannot judge whether the answer is
  reasonable.
• The progression towards a written method is
  crucial, since it is based on steps which are
  done mentally and which need to be secured first.
• A structured approach to developing written
  methods relies on
• Apparatus
• Apparatus picture
• Apparatus picture recording
• Picture recording
• Recording
• Developing a common whole school approach
  enhances pupils confidence and aids continuity
  across the school.
• It can be hard for a pupil to hold all the steps
  of a calculation in their heads, so informal
  jottings should be encouraged at all times.
• The transition between stages should not be
  hurried as not all children will be ready to move
  on to the next stage at the same time, therefore
  the progression in this document is outlined in
  stages. Previous stages may need to be revisited
  to consolidate understanding when introducing a
  new strategy.

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• Progression in Teaching Addition
• Mental Skills
• Recognise the size and position of numbers
• Count on in ones and tens
• Know number bonds to 10 and 20
• Add multiples of 10 to any number
• Partition and recombine numbers
• Bridge through 10
• Models and Images

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Recognise numbers 0 to 10
Cardinality Knowing that 6 objects is the
number 6. Count reliably up to 10 everyday
objects
Find one more than a number
Count in ones and tens
Count all
and
makes 5
Count on (keeping number in your head / hiding
the first pattern)
3 2 5
Begin to use the and signs to record mental
calculations in a number sentence
6 4 10
Know doubles of numbers and their corresponding
halves
5 5 10

5
Know by heart all pairs of numbers with a total
of 20
Know that addition can be done in any order
3
3 5
Put the biggest number first and count on
5
8
Begin to partition and recombine numbers in order
to add
6
15 1 16
Know which digit changes when adding 1s or 10s to
any number
15
16
15 10 25
25
15 20 35
15
25
35
Adding two two-digit numbers (without
bridging) Counting in tens and ones
15 13 28 15 10 3 28
Adding two two-digit numbers (bridging through
tens boundary) Using a number line
48 30 6 84
48 36 84
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Adding two two-digit numbers (bridging through
tens boundary) Using place value cards and place
value apparatus to partition numbers and recombine
48 36 14 (8 6)
70 (40 30) 84
48 36 84
Remember to add the least significant digits
(e.g. units in this case) first
It is important that the children have a good
understanding of place value and partitioning
using concrete resources and visual images to
support calculations. The expanded method enables
children to see what happens to numbers in the
standard written method.
48 36 84
T U 40 8 30
6 80 4 10
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Standard written method The previous stages
reinforce what happens to the numbers when they
are added together using more formal written
methods.
225 156 381 200 20 5
100 50 6 300 80 1
10
2 2 5 1 5 6 3 8 1 1
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• Progression in Teaching Subtraction
• Mental Skills
• Recognise the size and position of numbers
• Count back in ones and tens
• Know number facts for all numbers to 20
• Subtract multiples of 10 from any number
• Partition and recombine numbers (only partition
  the number to be subtracted)
• Bridge through 10
• Models and Images
• Counting apparatus
• Place value apparatus

-
10
Ten green bottles hanging on the wall
Five fat sausages frying in a pan
Begin to count backwards in familiar contexts
such as number rhymes or stories
Continue to count back in ones from any given
number
Three teddies take away two teddies leaves one
teddy
Take away
Find one less than a number
Count back in tens
If I take away four shells there are six left
Count backwards along a number line to take
away
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Maria had six sweets and she ate
four. How many did she have left?
Begin to use the and signs to record mental
calculations in a number sentence
6 - 4 2
Know by heart subtraction facts for numbers up to
10 and 20
Begin to find the difference
40 27 13
Find the difference using the number line
10
1
1
1
27
40
30
28
29
12
Find the difference using the number line
40 27 13
10
3
27
30
40
74 - 27 47
Find the difference using the number line
Extend to 3 digit numbers and decimals
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column subtraction (without exchanging)
40 3 - 30 2 10 1

• 43 32 11
• 4 3
• 3 2
• 1 1

Extend to 3 digit numbers without exchanging.
Expanded method It is important that the children
have a good understanding of place value and
partitioning using concrete resources and visual
images to support calculations. The expanded
method enables children to see what happens to
numbers in the standard written method.
column subtraction (with exchanging)
Standard written method The previous stages
reinforce what happens to numbers when they are
subtracted using more formal written methods. It
is important that the children have a good
understanding of place value and partitioning.
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Progression in Teaching Multiplication Mental
Skills Recognise the size and position of
numbers Count on in different steps 2s, 5s, 10s
Double numbers up to 10 Recognise multiplication
as repeated addition Quick recall of
multiplication facts Use known facts to derive
associated facts Multiplying by 10, 100, 1000 and
understanding the effect Multiplying by multiples
of 10 Models and Images Counting
apparatus Place value apparatus Arrays 100
squares Number tracks Numbered number
lines Marked but unnumbered lines Empty number
lines. Multiplication squares Counting stick Bead
strings Models and Images charts ITPs
Multiplication grid, Number Dials, Multiplication
Facts Vocabulary lots of groups
of times multiply multiplication multiple product
once, twice, three times array, row,
column double repeated addition
x
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Count in tens from zero
Count in twos from zero
Count in fives from zero
5
0
10
15
20
25
30
Know doubles and corresponding halves
Know multiplication tables to 10 x 10
x 5
6 x 5 30
2 x 5 10
3 x 5 15
8 x 5 40
Use known facts to work out new ones
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Understand multiplication as repeated addition
2 2 2 2 8 4 x 2 8 2 multiplied by 4 4
lots of 2
Understand multiplication as an array
Understand how to represent arrays on a number
line
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Use place value apparatus to support the
multiplication of U x TU
13 x 4 52
Use place value apparatus to support the
multiplication of U x TU alongside the grid
method
13 x 4 52
40 12 52
Use place value apparatus to represent the
multiplication of U x TU alongside the grid
method
23 x 4 92
80 12 92
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Multiplying TU x TU
33 x 14 462
Use same method to multiply HTU X U
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Progression in Teaching Division Mental
Skills Recognise the size and position of
numbers Count back in different steps 2s, 5s,
10s Halve numbers to 20 Recognise division as
repeated subtraction Quick recall of division
facts Use known facts to derive associated
facts Divide by 10, 100, 1000 and understanding
the effect Divide by multiples of 10 Models
and Images Counting apparatus Arrays 100
squares Number tracks Numbered number
lines Marked but unnumbered lines Empty number
lines. Multiplication squares Models and Images
charts ITPs Multiplication grid, Number Dials,
Grouping, Remainders Vocabulary lots
of groups of share group halve half divide divisio
n divided by remainder factor quotient divisible

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Count on in tens
Count on in twos
Count on in fives
Know halves
Half of 6 is 3 ½ of 6 3
If 2 x 10 20 then 20 ? 10 2 20 ? 2 10
Use known multiplication facts to work out
corresponding division facts
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Understand division as sharing
Understand division as grouping
12 divided into groups of 3 gives 4 groups 12 ? 3
4
12 divided into groups of 4 gives 3 groups 12 ? 4
3
Reinforce division as grouping through the use of
arrays
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18 ? 4 4 r 2
18 divided into groups of 3 18 ? 3 6
Represent groups for division on a number line
using apparatus alongside the line
1 x 3
1 x 3
1 x 3
1 x 3
1 x 3
1 x 3
3
0
18
6
9
12
15
Remember to ask the question how many groups
of 3 in 18?
Introduce remainders
1 x 4
r 2
1 x 4
1 x 4
1 x 4
0
4
8
12
16
18
How many groups of 4s in 18? There are 4 groups
of 4 with 2 left over.
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Children need to see that as the numbers get
larger, chunking groups is the more efficient
method.
65 ? 5 13
50
15
10 x 5
3 x 5
0
50
65
How many groups of 5 in 65? There are 13 groups
of 5 in 65?