Adwick Primary School’s Written Calculations Policy 2013-2014 Addition Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Pictures/ marks; Objects (fingers, cubes, bead strings/ bars etc).
Number track/ 100 square then moving onto a prepared number line (The transition from the use of a number track to a number line is important and needs consideration).
Empty number lines
Partitioning (children need to be secure with place value in order to move onto this method).
Columns – Expanded method
Columns – Compact method
Pictures/ marks
Number Track/ 100 square using counters
Partitioning
Write the numbers in columns, adding the units first:
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, etc.
Use Number Track/ 100 square and counters to solve simple calculations in addition and support recording of written calculations.
First counting on in tens and ones (children need to be secure with partitioning before using this method).
In this method, recording is reduced further. Carry digits are recorded below the line, using the words ‘carry ten’ or ‘carry one hundred’, not ‘carry one’. Column addition remains efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable.
1
2
3
4
5
6
7
8
9
34 + 23 = 57
10
Number track demonstrating jumps
Then adding the units in one jump (by using the known fact 4 + 3 = 7).
6 add 3
34 + 23 = 57 +1
+1
Objects
Followed by adding the tens in one jump and the units in one jump.
Show me 3 1
2
3
4
5
6
7
8
9
34 + 23 = 57
Prepared number line demonstrating jumps
Say together: 3 and 2 is 5.
Children should use bead strings/ bars to support their understanding.
8 add 2 Record as: 8 + 2 = 10
Children then begin to use a prepared number line to support their own calculations using the number line to count on in ones, then progressing to counting on in tens and ones.
Bridging through ten can help children become more efficient (bead strings or bead bars can be used to demonstrate bridging through ten). 37 + 15 = 52
2
3
4
5
6
7 8
7
2
4
1
1
(7 + 4)
8
0
(60 + 20)
9
1
Using this method, children will: add several numbers with different numbers of digits begin to add two or more decimal numbers (know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g. 401.2 + 26.85 + 0.71)
367 + 24 13 + 110 = 123
3
+
2
7
9 10 11 12 13 14 15
6
2
5
6
4 7
8 3
+
4
1 8
1 0
(7 + 4) (60 + 20)
3
0
0
(300 + 0)
3
9
1
247 + 76 247 + 76 = (7 + 6) + (40 + 70) + (200 + 0) 7 + 6 = 13 40 + 70 = 110 200 + 0 = 200
6
+
+
1
7
.
6
0
1
2
.
0
7
0
.
0
7
0
.
6
0
9
.
0
0
2
0
.
0
0
2
9
.
6
7
49 + 73 = 122
Decimals 12.8 + 10.5 = 23.3 + 10
+ 0.2
7
6
4
8
1
4
8
6
9
1
3
4
1
1
1
22.8
23
(0 + 0.07) (0.6 + 0)
Subtraction
4
5 2
1
1
6
5
8
4
+
5
8
4
8
1
2
4
3
2
1
1
1
6
(7 + 2)
4
2
2
0
4
3
2
8
5
7
8
6 3
(10 + 10)
+
4
6
8
1
1
9
4
4
1
2
1
+ 0.3 23.3
7
8 5
Decimals
1 12.8
6
Children should extend the carrying method with any number of digits.
17.6 + 12.07 Children will continue to use empty number lines with increasingly large numbers, including compensation where appropriate.
3 +
1
13 + 110 + 200 = 323 +1 +1 +1 +1 +1
1
6
Decimals
8 add 5 Record as: 8 + 5 = 13
0
47 + 76
____________________ _
How many altogether? (count
Record as: III + II is 5 leading to 3 + 2 = 5
+
10
Show me 2’
1,2,3… 1,2… 1,2,3,4,5.)
67 + 24
47 + 76 = (7 + 6) + (40 + 70) 7 + 6 = 13 40 + 70 = 110
Record as: 6 + 3 = 9 +1
Add the units first and then the tens etc to form partial calculations and then add these formal calculations using a method that the child can add with.
Stage 1
Stage 2
Stage 3
Stage 4
Pictures/ marks; Objects (fingers, cubes, bead strings/ bars etc).
Number track/ 100 square then moving onto a prepared number line (The transition from the use of a number track to a number line is important and needs consideration).
Empty number lines
Partitioning (children need to be secure with place value in order to move onto this method).
Pictures/ marks
Number Track/ 100 square to take away
Counting back First counting back in tens and ones (children need to be secure with partitioning before using this method).
Partitioning
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 etc.
Taking away 4 take away 2 Record as: 4 – 2 = 2 1
2
3
4
5
6
7
8
9
10
Using a number track/ 100 square to count back.
Then subtracting the units in one jump (by using the known fact 7 – 3 = 4).
Number track demonstrating jumps
Objects
9 take away 3 Record as: 9 – 3 = 6
Show me 5
-1
-1
-1
Followed by subtracting the tens in one jump and the units in one jump.
Take away 2 1
2
3
4
5
6
7
8
9
Subtraction can be recorded using partitioning to write equivalent calculations that can be carried out mentally. For 97 – 25, this involves partitioning the 25 into5 and 20, and then subtracting from 97 the 5 and the 20 in turn.
Stage 5
Stage 6
Columns – Expanded method
Columns – Compact method
Partitioning without decomposition Start by subtracting the ones, then the tens, then the hundreds. Refer to subtracting the tens, for example, by saying ‘sixty take away forty’, not ‘six take away four’.
-
60
3
200
40
1
300
20
2
-
Prepared number line demonstrating jumps Children then begin to use a prepared number line to support their own calculations using the number line to count back in ones, then progressing to counting back in tens and ones.
Counting on If the numbers involved in the calculation are close together or near to multiples of 10, 100 etc, it can be more efficient to count on from the smallest number to the largest number.
For 297 – 25, this involves partitioning the 25 into 5 and 20, and then subtracting from 297 the 5 and the 20 in turn.
78 – 56 = 22 (56 count on 22)
297 – 25
+10
+10
+1
+1
15 take away 4 Record as: 15 – 4=11 6 take away 2 Record as: 6 – 2 = 4
-1
0
1
2
3
4
5
6
7 8
-1 -1
6
3
2
4
1
3
2
2
6
1
7
4
2
7
4
7
10
Say together:‘5 take away 2 is 3.
Children should use bead strings/ bars to support their understanding.
5
TU – TU 500
How many now? (count 1,2,3.) Record as: IIIII - II is 3 leading to 5 –2=3
-
Decomposition
563 – 241
97 - 25 97 – 25 = 97 - 5 – 20 97 – 5 = 92 92 – 20 = 72
In this method, recording is reduced further. Column subtraction remains efficient when used with larger whole numbers and decimals. Subtraction in columns may require decomposition.
-1
9 10 11 12 13 14 15
56 66 76 77 78 Then help children to become more efficient with counting on by: Counting on the tens in one jump and the units in one jump Counting on the units in one jump Bridging through ten
297 – 25 = 297 – 5 – 20 297 – 5 = 292 292 – 20 = 272
=
322
HTU - HTU Partitioning and decomposition 74 – 27 60
-
-
14
70
4
20
7
40
7
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.
14 5
1 4
2
8
6
4
6
8
ThHTU - ThHTU =
47
741 – 367 -
Finding the difference
6 7
5
13 1
6
4
6
7
2
6
8
4
3
7
8
3
130
The ‘counting on’ method can be used with Decimals where no more than three columns are required as it becomes less efficient when more than three columns are needed.
1
-
800
30
1
700
40
1
300
60
7
500
70
4
Decimals
=
3 4
574 -
0
1
2
3
4
5
6 7
8
9 10 11 12 13 14 15
3
Multiplication
9
0
10 1
.
11 2
1 0
2
6
.
8
5
7
4
.
3
5
1
Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Pictures/ marks; Objects (fingers, cubes, bead strings/ bars etc).
Prepared number line
Empty number lines
Partitioning (children need to be secure with place value in order to move onto this method).
Columns – Expanded method
Columns – Compact method (Short multiplication)
Pictures/ marks
Children will develop their use of repeated addition. Initially, these should be multiples of 10s, 5s, and 2s – numbers with which the children are more familiar.
Using an empty number line, children will continue to develop their use of repeated addition.
Partitioning
The next step is to represent the method of recording in a column format, but showing the working. Draw attention to the links with the grid method.
The recording is reduced further, with carry digits recorded below the line.
Children will experience equal groups of objects. They will count in 2s and 10s and begin to count in 5s. They will work on practical problem solving activities involving equal sets or groups.
Repeated addition using a prepared number line Repeated addition can be shown easily on a prepared number line:
Repeated addition using an empty number line
5
5
0
3 times 4 is 4 + 4 + 4 = 12 or 3 lots of 4 or 3 groups of 4 or 3 x4
Children should use bead strings/ bars to support their understanding.
1
2
3
4
5
6
7
8
Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 x 7 is ‘8 x 7’ and then ’30 x 7’, not ‘3 x 7’, although the relationship 3 x 7 should be stressed.
38 x 5 4x6 6 + 6 + 6 + 6 = 24
3 times 5 or 3 lots of 5 or 3 groups of 5 Record as: 3x5=5+5+5 5 + 5 + 5 = 15 Objects
Multiply the units first and then the tens etc to form partial calculations and then add these formal calculations using a method that the child can add with.
+6
+6
38 x 5 = (8 x 5) + (30 x 5) 8 x 5 = 40 30 x 5 = 150
+6
+6
40 + 150 = 190
9 10 11 12 13 14 15
0
6
12
24
Arrays Children should be able to model a multiplication calculation using an array. This knowledge will support with the development of the grid method.
Commutativity Children should know that 3 x 5 has the same answer as 5 x 3. This can also be shown on the number line.
18
492 x 3
38 x 7
4.92 x 3 = (0.02 x 3) + (0.9 x3) + (4 x 3) 002 x 3 = 006 090 x 3 = 270 400 x 3 = 1200
3 X
0.06 + 2.70 + 12 = 14.76
7
20
2
3
140
21
6
6
5
Long multiplication 2
8
6
2
9
5
7
4
(9 x 286)
5
7
2
0
(20 x 286)
8
2
9
4
6
(8 x 7 = 56)
x
1
0
(30 x 7 = 210)
2
23 x 7 x
7 2
5
Grid Method 3 x 5 = 15
8
8 7
2
3 x
Children who are already secure with multiplication for TU × U and TU × TU should have little difficulty in using the same method for HTU × TU.
TU x U
5
Short multiplication
6
6
= 161
1
HTU x TU
5
5
286 x 29
5 5 x 3 = 15
2
72 x 38 3 lots of 5 or 3 groups of 5 Record as: 3 x 5 = 15
x
70
2
30
2100
60
8
560
x
16
= 2736 1
8
6
2
9
5
4
7
2
0
8
0
0
1
2
0
1
6
0
0
Multiplying by a Decimal number
4
0
0
0
27 x 3.2
8
2
9
4
x
20
7
3
60
21
0.2
4
1.4
Decimal by a whole number 4
(9 x 6 = 54) (9 x 80 = 720) (9 x 200 = 1800) (20 x 6 = 120) (20 x 80 = 1600) (20 x 200 = 4000)
2
**Note carries should be recorded = 86.4 beneath.
x
3
1
4 2
.
9
2
.
7
6
Division Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Stage 6
Pictures/ marks; Objects (fingers, cubes, bead strings/ bars etc).
Prepared number line
Empty number lines
Partitioning (children need to be secure with place value in order to move onto this method).
Columns – Expanded method
Columns – Compact method (Short division)
Pictures/ marks
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, and 2s – numbers with which the children are more familiar.
Using an empty number line, children will continue to develop their use of repeated subtraction to be able to subtract multiples of the divisor.
Partitioning
Chunking is based on subtracting multiples of the divisor, or ‘chunks’. Initially children subtract several chunks, but with practice they should look for the biggest multiples of the divisor that they can find to subtract (see condensed chunking example blow)
‘Short’ division of TU ÷ U can be introduced as a more compact recording of the mental method of partitioning.
Children will understand equal groups and share items out in play and problem solving. They will count in 2s and 10s and later in 5s.
Repeated subtraction using a prepared number line Repeated subtraction can be shown easily on a prepared number line:
Repeated subtraction using an empty number line 24 ÷ 4 24 - 4 - 4 - 4 – 4 – 4 - 4 (subtracted 6 equal groups of 4 to reach 0) 24 ÷ 4 = 6
15 divided by 5 or 15 shared by 5 or 15 grouped into 5s Record as: 15 ÷ 5 = 3
Objects Sharing equally 6 sweets shared between 2 people, how many do they each get?
-5
0
1
2
-5
3
4
5
6
7
-5
8
9
10 11 12 13 14 15
-4 0
-4 4
-4 8
12
-4
-4
16
20
24
13 ÷ 4 = 3 r1
155 ÷ 5
0
1
-4
-4
5
9
13
72 ÷ 5 = 14 r2
155 5 = (5 5) + (50 5) + (100 5) 55=1 50 5 = 10 100 5 = 20
1 + 10 + 20 = 31 Children should use bead strings/ bars to support their understanding. 4 4
Moving onto:
2 3
136 ÷ 2 = (6 ÷ 2) + (30 ÷ 2) + (100 ÷ 2) 6÷2=3 30 ÷ 2 = 15 100 ÷ 2 = 50 3 + 15 + 50 = 68
-4
The short division method is recorded like this:
136 ÷ 2
Children should also move onto calculations involving remainders.
r1 Grouping or repeated subtraction There are 6 sweets, how many people can have 2 sweets each?
-4
Divide the units first and then the tens etc to form partial calculations and then add these formal calculations using a method that the child can add with.
**This method only works
7 2
8
1
The carry digit ‘2’ represents the 2 tens that have been exchanged for 20 ones. In the first recording above it is written in front of the 1 to show that 21 is to be divided by 3. Condensed chunking The key to the efficiency of chunking lies in the estimate that is made before the chunking starts. Estimating has two purposes when doing a division: to help to choose a starting point for the division; to check the answer after the calculation.
‘Short’ division of HTU ÷ U can be introduced as an alternative, more compact recording. No chunking is involved since the links are to partitioning, not repeated subtraction.
9 3
2 9
7 2
1
in a simple form for dividing by 2, 5 or 10.
8 shared by 2 Record as: 8 ÷ 2 = 4 or 2 2 2
8 grouped into 2s Record as: 8 ÷ 2 = 4
Counting on
2
When children become competent with inverse then they can use repeated addition to solve division calculations as it is easier to count on than it is to count back.
87.5 ÷ 7
4x6 6 + 6 + 6 + 6 = 24
+6
0
+6
6
7 -
+6
12
The answer 32 (with a remainder of 4) lies between 30 and 40, as predicted.
+6
18
24
8 7 1 1
7 0 7 4 3 3 0
. . . . . . .
5 0 5 0 5 5 0
Answer = 12.5
Children will have a range of calculation methods, mental and written; selection will depend upon the numbers involved.
(10 X 7) (2 X 7) (0.5 X 7)
The carry digit ‘2’ represents the 2 tens that have been exchanged for 20 ones. In the first recording above it is written in front of the 1 to show that a total of 21 ones are to be divided by 3. The 97 written above the line represents the answer.
Children should not be made to go onto the next stage if: 1) 2)
they are not ready they are not confident
* Some children may not grasp certain stages. It is important that they find a method that suits them and is appropriate for the calculation involved. Children should be encouraged to:  approximate their answers before calculating;  check their answers after calculation using an appropriate strategy;  consider if a mental calculation would be appropriate before using written methods. * KS1 Level 3 SATs guidelines stipulate that children cannot use equipment, therefore this needs consideration when preparing children who will be taking the Level 3 paper in terms of which written calculation methods they are encouraged to use (stage 3 onwards).