Many parents and carers ask us how they can help their child with maths. One of the
best things you can do with your children is talk to them about maths. Sometimes it is even
more important that you listen and let them explain what they are doing and why. This
booklet is designed to help you to understand some of the methods of calculation that they
may be working on in class.
In Key Stage 1 children often draw pictures and diagrams to help them to understand their
workings. A lot of emphasis is placed on asking the children to explain how they work things
out; how they arrived at a particular answer. We acknowledge the childrens’ ability to use
different approaches to the same sum, for example, when adding 18 and 9 some children
will count on from 18 where as others may add on 10 then take away 1.
In the Key Stage 2 years the children will do more work with written sums. If you don’t
recognise these methods please don’t insist that your child carry out calculations in the way
that you do them. There are often several different ways of calculating and each child will
find the method that he/she feels most comfortable with and leads to a consistent level of
accuracy.
This booklet explains the expanded informal methods which show the children the
reasoning behind the steps involved in calculations, before moving onto the more common
compact, standard methods.
1
Children working on the Year 3 objectives will use informal pencil and paper methods
(jottings) to support, record or explain their calculations. These methods build upon
existing mental strategies.
Most children will set sums out horizontally, like this
86 + 57 =
and work them out using methods such as;
a) 86 + 57 = 86 + 50 + 7
= 136 + 7 = 143
b) 86 + 57 = (80 + 50) + (6 +7)
= 130 + 13
= 143
+50
+4
+3
c)
_____________________________
86
136
140
143
These are preparation for efficient, standard methods where calculations are set out in
columns; units line up under units, tens under tens and so on.
46
+ 27
60 (40 + 20, adding tens first)
+ 13 ( 6 + 7, then adding units)
73
or
46
+ 27
13
+ 60
73
2
(adding units first)
( then adding tens)
256
+ 85
11 (units)
+ 130 (tens)
200 (hundreds)
341
Children working on the Year 4, 5 and 6 objectives will use pencil and paper methods to
support, record or explain their calculations, achieving consistent accuracy.
When setting sums out vertically the children will continue to line up tens under tens, units
under units and so on. Examples of some of the methods are as follows;
a) adding the most significant digits first, mentally, from the top,
working on Yr 4 objectives
625
+ 48
600
+ 60
13
673
(adding hundreds)
(adding tens)
(adding units)
Year 5 objectives
587
+ 475
900 (adding hundreds)
+ 150 (adding tens)
12 (adding units)
1062
Year 6 objectives
7648
+ 1486
8000
+1000
120
14
9134
(add thousands)
(add hundreds)
(add tens)
(add units)
b) compensation ( add too much, take off)
Year 4 objectives
754
+ 86
854
- 14
840
(754 + 100)
(86 to 100 is 14)
(86 is rounded up to 100)
Year 5 objectives
654
+ 286
954 (654 + 300)
- 14 (286 to 300)
940
(286 is rounded up to 300)
3
Year 6 objectives
N/A
c) standard written methods, adding the least significant digits first, using “carrying”
Year 4 objectives
358
+ 73
11 (units)
+ 120 (tens)
300 (hundreds)
431
Year 5 objectives
358
+ 73
( leading to)
587
+ 475
1062
431
1
11
Year 6 objectives
7648
+ 1486
9134
111
6584
+5848
12 4 3 2
1 1 1
d) extend to decimals, using similar methods to those above, ensuring that the decimal
points line up with each other.
4
Children working on the Year 3 objectives will use informal pencil and paper methods (jottings) to
support, record or explain partial mental methods, building on existing strategies.
Most children will set sums out horizontally, like this,
such as;
a)
84 – 56 and work them out using methods
counting up from the smaller to the larger number (complimentary addition)
1) 84 - 56
+4
+20
+4
or
_______________________________
56
60
80
84
becomes 56 + 4 + 20 + 4 = 84
4 + 20 + 4 = 28
+4
+40
+300
+83
_________________________________________
356
360
400
700
783
2) 783 – 356
b) compensation (take too much, add back)
1) 84 – 56
=
84 – 60 + 4
= 28
4
+ 40
300
83
427
( 56 is rounded up to 60 )
-60
___+4________________________
24
28
84
2) 783 – 356 = 783 – 400 + 44
= 383 + 44
= 427
c) decomposition, recording calculations in preparation for an efficient standard
method
81
- 57
=
80 + 1
- 50 + 7
=
70 + 11
(adjust from T to U)
- 50 + 7
20 + 4 = 24
5
Children working on the Year 4, 5 and 6 objectives will use pencil and paper methods to
support, record or explain their calculations, achieving consistent accuracy.
When setting sums out vertically they will continue to line up tens under tens, units under units and
so on. Examples of some of the methods are as follows;
a) counting up ( complimentary addition)
working on the Year
objectives
754
- 86
+4
+10
+ 600
+50
+4
668
4
to 86 to make 90
to 90 to make 100
to make 700
to make 750
to make 754
Year 5
objectives
754
- 286
Year 6
objectives
6467
-2684
+ 14 to 286 to make 300
+400 to make 700
+ 54 to make 754
468
16 ( count up, 2684 to 2700)
+ 300 (count up, 2700 to 3000)
3467 (count up, 3000 to 6467)
3783
b) compensation, (take too much, add back)
working on the Yr
754
- 86
654
+ 14
668
4
objectives
(round up to 100)
(754 - 100)
(86 to 100 = 14)
Year 5
objectives
754
- 286 (round up to 300)
454 (754 - 300)
+ 14 (286 to 300 = 14)
468
Year 6
6467
- 2684 (round up to 3000)
3467 (6467 – 3000)
+ 316 (2684 to 3000)
3783
1
6
objectives
c) standard written methods, decomposition
working on the objectives for
754 =
- 86
Year 4 and…….
700 + 50 + 4
-
Year 5
leading to
leading to
(Yr
86 + 6
4
=
5)
1
700 + 40 + 1 4 (adjust from T to U)
7 5 4
80 + 6
- 8 6
d) extend to decimals, ensuring that the decimal points line up with each other.
=
4 1
600+ 140 + 14 (adjust from H to T)
80 + 6
600+ 60 + 8 = 668
7 5 4
- 8 6
6 14 14
754
-86
668
d) extend to decimals, ensuring that the decimal points line up with each other.
7
and
Year 6
.
5 3 1
6 4 6 7
- 2 6 8 4
3 7 8 3
Children working on the Year 3 objectives will use informal pencil and paper methods
(jottings) to support, record or explain their calculations. These methods build upon
existing mental strategies.
Multiplication
The children will need to know the 2, 5, and 10 times tables and have a working knowledge of
the 3 and 4 times tables by the end of the academic year.
They will be working on such problems as “If there are 10 gingerbread men in 1 packet, how
many will there be in 3 packets?”
Division
The knowledge of the above multiplication tables is necessary for the objectives relating to
division, along with a knowledge of doubles and halves. The children will be working on such
problems as “How many lots of 3 go into 9?” and “What is 80 divided by 10?”.
They will need to recognise the – sign and use the vocabulary related to division, such as
“share”.
8
Children working on the Year 4, 5 and 6 objectives will use pencil and paper methods to
support, record or explain their calculations, achieving consistent accuracy. They will discuss,
explain and compare methods.
Children working on the Year
Mental method, partitioning.
4
objectives;
Grid layout, expanded working. Standard written methods
38 x 7 = (30 x 7) + (8 x 7)
Children working on the Year
Grid layout, larger numbers
5
X
30
7
210
Short multiplication TU x U
Eg 23 x 7. Estimate 20x10=200
8
23
leading to
x 7
140 (20x7)
21 (3x7)
161
56 = 266
objectives
Example: TU x TU
56 x 27
Estimate: 60 x 30 =,1800
56 x 27 = (50 + 6) x (20 + 7)
Example: HTU x U 346 x 9
Estimate: 350 x 10 = 3500
346 x 9 = (300 + 40 + 6) x 9
X
20
7
X 300
9 2700
50
1000
350
6
120 =
42 =
1120
+ 392
1512
40
360
6
54 = 3114
Standard written methods
Short multiplication ; HTU x U
Example: 346 x 9
Estimate 350 x 10 = 3500
346
x 9
2700
+ 3 60
54
3 1 14
leading to
(300 x 9)
(40 x9)
(6 x 9)
Long multiplication: TU xTU
Example: 72 x 38
Estimate: 70 x 40 = 2800
346
x 9
3 1 14
45
72
x 38
2160 (30x72)
+
576 (8x72)
2736
1
9
Contd.
23
x7
161
2
Children working on the Year 5 objectives. (contd)
Extend to simple decimals with one decimal place
Multiply by a single digit, estimating
first.
Example 4.9 x 3
Estimate 5 x 3 = 15
4.0 x 3 = 12.0
0.9 x 3 = 2.7
14.7
Children working on the Year 6
objectives
Informal written methods; grid method
Example: Th H T U x U
Example: 4346 x 8
Estimate: 4500 x 10 = 45000
Example: HTU x TU
Example: 372 x 24
Estimate: 400 x 20
X
8
X 300
70 2
20 6000 1400 40 = 7440
4 1200 280 8 = + 1488
8928
4000
300 40 6
32000 2400 320 48 = 34768
Standard written methods
Short multiplication: Th H T U x U
Example: 4346 x 8
Estimate: 4500 x 10 = 45000
Long multiplication: HTU x TU
Example: 352 x 27
Estimate: 350 x 30 = 10500
4346
X
8
32000
2400
+ 320
48
34768
352
X 27
7040
+ 2464
9504
leading to
(4000 x 8)
(300 x 8)
( 40 x 8 )
(6x8)
4346
x
8
34 7 6 8
23 4
(352 x 20)
(352 x 7)
1
10
Year 6 contd on next page.
Children working on the Year 6 objectives. (contd)
Extend to decimals with up to two decimal places
Multiply by a single digit, estimating first. Decimal points should line up under each other.
Example: 4.92 x 3
Estimate: 5 x 3 = 15
4.92
X 3
14.76
2
Begin to extend to multiplying by two-digit numbers
Example; 4.92 x 73
Estimate: 5 x 70 = 350
4.92
X 73
344.40
14.76
359.16
(4.92 x 70)
(4.92 x3)
11
Children working on the Year 4, 5 and 6 objectives will use pencil and paper methods to
support, record or explain their calculations, achieving consistent accuracy. They will discuss,
explain and compare methods.
Working at the Year
4 level
a) informal written method, using multiples of the divisor
Example: 72 ÷ 5
Estimate: lies between 50 ÷ 5 = 10 and 100 ÷ 5 = 20
72 ÷ 5
Or:
72 ÷ 5
= (50 + 22) - 5
= 10 + 4 remainder 2 or 14 remainder 2
= 72
- 50
22
-20
2
Answer
(10 x 5)
( 4 x 5)
14
14 remainder 2
b) standard written method; short division TU ÷ U
Example: 96 ÷ 6
Estimate: 100 ÷ 5 = 20
____
6) 96
- 60 (10 x 6)
36
-36 (6 x 6 )
0
Answer
16
12
Working on the Year 5 objectives
a) informal written method, using multiples of the divisor
Example: 256 ÷ 7
Estimate: lies between 210 ÷ 7 = 30 and 280 ÷ 7 = 40
256
- 70
186
-140
46
- 42
4
(10 x 7)
(20 x 7)
( 6 x 7)
36
Answer 36 remainder 4
c) Standard written methods: short division HTU ÷ U
Example: 196 ÷ 6
Estimate: 200 ÷ 5 = 40
______
6) 196
-180 (30 x 6)
16
- 12 ( 2 x 6)
4 32
leading to
32 R 4
6) 196
- 18
16
- 12
4
Answer 32 R 4
13
Working on the Year 6 objectives;
a) informal written method, using multiples of the divisor
Example; 977 ÷36
Estimate: 1000 ÷ 40 = 25
977
-360
617
-360
257
-180
77
72
5
(10 x 36)
(10 x 36)
( 5 x 36)
( 2 x36)
27
Answer 27 r 5
b) standard written method: long division
Example: 972 ÷ 36
Estimate: 1000 ÷ 40 = 25
____
36) 972
- 720
252
- 252
0
leading to
(20 x 36)
(7 x 36)
27
27
36) 972
-72
25 2
-25 2
0
Answer 27
c) extend to decimals with up to two decimal places
Example: 87. 5 ÷ 7
Estimate: 80 ÷ 8 = 10
______
7) 87.5
-70.0
(10 x 7)
17.5
-14.0
(2 x 7)
3.5
- 3.5 (0.5 x 7)
0.0 12.5
answer 12.5
14