Updated October 2012
Calculation Progression
for
Addition, Subtraction,
Multiplication and Division
This is Hillside’s Calculation Policy. These documents show the methods taught
throughout the school and the progression of the four operations. Please use the weekly
Home Learning or Maths Facts to gauge which stage your child is at or ask your child’s
class teacher.
It is important that we work together to use the same calculation methods in order that we
don’t confuse the children.




By the end of Year 6, children will have a range of calculation methods.
Selection will depend upon the numbers involved.
Children should not be made to go onto the next stage if:
o they are not ready
o they are not confident
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.
These notes show the stages in building up to a compact, efficient method for addition.
Our aim is that children use mental methods when appropriate but for calculations that
they cannot do in their heads they choose an appropriate written method which they can
use accurately and with confidence. Time must be taken to build up to the most efficient
method to ensure complete understanding at each stage.
Counting on from a number to find the total
Adding groups together (objects or pictures of objects). Use
moveable objects when finding totals. Touch and align each
object as it is counted.
Use fingers for calculations to 10.
Count first group, start count from first groups total when
counting second group.
Counting on from the biggest number
Number Bonds to 10 and 20
Adding two groups together to find the total.
Counting on in ones on a number line/hundred square
Biggest number in head and counting on
Jane had 3 bears. She was given 5 more. How many does she
have now?
To support learning of number facts using a variety of visual
resources
Addition by adding on in multiples of tens (by jumping
down) and ones (by jumping along) on a hundred square.
Number bonds to 10, 20 and pairs of multiples of tens to
100.
Addition by partitioning one number on an empty number
line
23 + 12 = 35
10 +2
+10
+2
___________________
23
33
35
Continued on next page
2
Addition as partitioning and recombining
42 + 36
40 + 2 + 30 + 6
70 + 8 = 78
Addition by partitioning one number on an empty number
line
34 + 35 =
+10
+10
+10
+5
__________________________________
34
44
54
64 69
Addition with chunking
34 + 35 =
+30
+5
____________________________________
34
64 69
Number bonds to 100
Addition by partitioning one number on an empty number
line:
124 + 145 = 124 + 100 +40 + 5
+100
124
+40
224
+5
264
269
Addition as partitioning and recombining:
133 + 158=
100 + 30 + 3 + 100 + 50 +8
200 + 80 + 11
= 291
Continued on next page
3
Stage 1 of column method:
Adding tens first then units up to 1000
(building on mental calculation strategies that add biggest
numbers first)
43
+54
90
7
97
Stage 2 of column method:
Expanded Addition
Adding units first, then 10’s, then 100’s
358
+ 33
11
80
300
391
Stage 3 of column method:
Compact Method
Adding units first then 10’s then 100’s
358
+ 33
391
1
Stage 3 of column method:
Compact method/formal written method
Extend to addition of 4 digit numbers
3587
+ 675
4262
11 1
Extend to addition of more than 2 numbers
671
98
468
1237
21
Continued on next page
4
Extend to 1 place decimals
72. 5
+54. 6
127. 1
1
Extend to 2 place decimals
£ 73.42
+£ 84.73
£158.15
1
Continue with compact method
3481.9
26.85
+ 0.71
3509.46
1
2
Addition with time on the number line method
+1 hour
+17mins
_______________________________________
9:23
10:23
10:40
5
These notes show the stages in building up to a compact, efficient method for subtraction.
Our aim is that children use mental methods when appropriate but for calculations that
they cannot do in their heads they choose an appropriate written method which they can
use accurately and with confidence. Time must be taken to build up to the most efficient
method to ensure complete understanding at each stage.
To use practical apparatus to take away objects
Using fingers to subtract
Subtraction stories to ten using pictures
Subtraction as counting back on a number line/hundred square
in ones
Subtraction facts
Finding the difference between two numbers e.g. 5 and 8
Subtraction as counting back in tens and ones on a number line
and a hundred square.
Subtraction as taking away on an empty number line
18-12=
-10
-2
_____________________________
6
8
18
Subtraction by partitioning
56 – 23 = 56 -20
= 36 – 3
= 33
Subtraction as taking away on an empty number line
163 -47 =116
Count back 47 from 163
-7
-10
-10
-10
-10
_______________________________________
116 123 133 143 153 163
Continued on next page
6
Working towards taking away 40 (4 x 10) to be more efficient
-7
-40
_______________________________________
116 123
163
Subtraction as finding the difference
113 – 87 =26
-3
-10
-10
-3
___________________________________
0
87 90
100
110 113
Finding the difference (counting up from the smaller number
strategy)
754 – 586 = 168
754
-586
4
10
100
50
4
168
(To make 590)
(To make 600)
(To make 700)
(To make 750)
(To make 754)
Finding the difference (counting up method)
£ 14. 24
-£ 8. 70
.30 (to make £ 9.00)
£5.00 (to make £14.00)
£0.24 (to make £14.24)
£ 5.54
Leading to compact/formal written method
£ 14. 24
-£ 8. 70
.30 (to make £ 9.00)
£5.24 (to make £14.24)
£5.54
Continued on next page
7
Partitioned compact method
754
-522
2 (4 – 2)
30 (50 – 20)
200 (700 – 500)
232
Leading to compact decomposition / exchanging
5 13 1
6476
-2684
3792
“If the Bigger number is on the
Bottom we Exchange”
We do not use the term ‘Borrow’
4 19 19 1
5001
- 567
4434
Extend to decimal numbers
3 91
£ 4.00
-£ 2.65
£1.35
Extend to mixed decimal places
3.6 – 2.04
51
3.60
- 2.04
1.56
8
These notes show the stages in the build-up to a compact, efficient method for
multiplication. Our aim is that children use mental methods when appropriate but for
calculations that they cannot do in their heads they choose an appropriate written method
which they can use accurately and with confidence. Time must be taken to build up to the
most efficient method to ensure complete understanding at each stage.
Counting in twos
Number rhymes in twos such as 2, 4, 6, 8
Hopping in twos
Counting on or back in 2, 5 and 10
Grouping in 2s, 5s and 10s
Counting in 2s, 5s and 10s
Use 100 square to see number patterns
Grouping objects
Understand multiplication as repeated addition
2x3=6
or 3 + 3 = 6
Understand multiplication as repeated addition:
With pictures
____
+
+
+ _____
+
_____ = ______
Understand multiplication as groups
2x3=6
Continued on next page
9
5 x 5 = 5 + 5 + 5 + 5 + 5 = 20
+5
+5
+5 +5 +5
____________________
0 5 10 15 20 25
Must know 2, 5 10 multiplication facts:
Understand multiplication as Arrays
7x5=
7 x 5 = 35
5 x 7 = 35
Should know all multiplication facts 1-10
Grid Method – TU x U
23 x 8
x 20
8 160
3
24
160
+ 24
184
Begin to mentally know multiples of 10s e.g. 30 x 7 = 210
10
Grid Method – HTU x U
253 x 6 =
x
6
200
1200
50
300
3
18
1200
300
+ 18
1518
Grid method – TU x TU
24 x 35 =
x
30
5
20
600
100
4
120
20
600
100
120
+ 20
840
Grid method – ThHTU x U
4732 x 5 =
x
5
4000 700
20000 3500
30
150
2
10
20000
3500
150
+
10
23660
Grid method – HTU x TU
156 x 43 =
x
40
3
100
50
4000 2000
300 150
6
240
18
4000
2000
240
300
150
+ 18
6708
11
Standard written method
Short multiplication – TU x U
34 x 6 =
2
34
x 6
204
Leading to HTU x U
836 x 4 =
12
836
x 4
3344
Short multiplication – U.t x U
8.6 x 8 =
4
8.6
x 8__
68.8
Leading to TU.t x U
56.9 x 7 =
46
56.9
x 7__
398.3
Long multiplication – TU x TU
36 x 47 =
3
36
x 47
252
1440
1692
2
12
These notes show the stages in the building up to a compact, efficient method for division.
Our aim is that children use mental methods when appropriate but for calculations that
they cannot do in their heads they choose an appropriate written method which they can
use accurately and with confidence. Time must be taken to build up to the most efficient
method to ensure complete understanding at each stage.
Counting in two’s
Number rhymes in two’s such as 2, 4, 6, 8
Hopping in twos on a number line
Counting on or back in 2, 5 and 10 on a hundred square
and fingers
Grouping in 2’s, 5’s and 10’s
Sharing objects between groups
Repeated subtraction on a number line / hundred square
Use grouping of objects
Sharing objects between groups
8 shared between 2
xx
xx
xx
xx
Answer = 4
8 objects grouped into 4 groups
xxxx
xxxx
Answer = 2
As grouping by repeated subtraction using a number
line:
Examples without the remainder
12 -3 -3 -3 -3 =
-3
-3
12÷3 = 4
-3
-3
__________________________________
0 1 2 3 4 5 6 7 8 9 10 11 12
13
Division as repeated subtraction
72 ÷ 5
-50
(10x5)
-20
(4x5)
_______________________________
0
2
22
72
72
-50 (10 x 5)
22
-20 (4 x 5)
Remainder
2
Answer = 14 r2
Chunking with TU ÷ U
56 ÷ 3 =
___
)56
-30 (10 x 3)
26
-15 (5 x 3)
11
- 9 (3 x 3)
2
Answer = 18 r2
Check = (18 x 3) + 2 = 56
Leading on to Chunking with HTU ÷ U
256 ÷ 7 =
____
)256
- 70 (10 x 7)
186
- 140 (20 x 7)
46
- 42 (6 x 7)
4
Answer = 36 r 4
Check = (36 x 7) + 4 = 256
14
Short division
284÷ 6
4 7 r2
6 )2 844
Remainder as a fraction
47 r2 is the same as
47
2
6
or 47
1
3
Remainders as Decimals
47 r2 is the same as
47.33
Long Division
284 ÷ 15
1 8 r14
15 )2 8 4
-15
134
-120
14
8362 ÷ 24
3 4 8 r10
24)8 3 6 2
-7 2
116
- 96
202
- 192
010
15