Teaching the 4 rules of Number
Standard written methods are based on steps which are done mentally. It is
important that the mental strategies required to tackle the written calculations are taught
and mastered first. Informal written methods can provide effective transition to standard
written methods, but they also enable children to visually see a mental process, for example
the use of an empty number line in addition.
It should be understood that we often vary the mental method we use to suit the
problem. For example working out 29 x 9 would involve a different strategy than 28 x 8.
The goal for written methods is to give the children a standard method that can be used for
all cases where a mental method is not appropriate.
The progress towards a standard written method should involve the following
stages:1. Establish mental methods, based on a good understanding of place value.
2. Present calculations in a horizontal format.
3. Show children how to set out written calculations vertically, initially using expanded
layouts.
4. As confidence grows refine this into a more compact method.
5. Extend to larger numbers and decimals.
 Always look for special cases that can be done mentally.
 Estimate first.
Generally children will be ready to meet standard written methods for addition and
subtraction when they have mastered the following: Addition to and subtraction from 20 mentally.
 Place value and partitioning numbers into hundreds, tens and ones.
 The laws of commutation and association i.e.
9 - 6 = 3 so 6 + 3 = 9
3 + 4 = 4 + 3 but 4 – 3 ≠ 3 – 4
4 + 13 = 17 therefore 14 + 3 = 17
 Mental addition of three single digit numbers.
 Mental addition and subtraction of any pair of two digit numbers.
Stages in Teaching Addition
1
1) Horizontal written expansion.
E.g.
77 + 46 = (70 + 40) + (7 + 6) or 77 + 46 = 110 + 13
(Larger number first) = 110 + 13
= 123
= 123
2) Use of number lines e.g.
+3
77
+3
80
+40
83
123
3) Vertical written expansion.
Children need to see this expansion to fully understand vertical addition, but it
may enough for them to watch a teacher demonstration.
47 = 40
7
76 = 70
6
______________
110
13 = 123
Use of base 10 apparatus is essential in explaining this stage.
Note
I.
II.
III.
Use of arrows reinforced by arrow cards. Less confusing than using + symbol,
especially when used later in subtraction etc.
Children with mixed laterality/dyslexic tendencies will find putting numbers in vertical
columns more helpful than the horizontal expansion.
Columns may initially be labelled
H T O
( note use of ones not units)
4) Vertically adding the least significant digit first in preparation for “carrying”
34
+ 52
47
+ 76
6
80
86
13
110
123
43
+ 25
Then
68
5) Vertically adding with carrying
47
This is the target stage for Y3 children up to 3 digits.
2
76
123
1
6) Use of larger numbers up to 4 digits, decimals with one decimal place and money.
(Year 4)
I. 368 + 496
368 = 300
496 = 400
700
60
90
150
8
6
14 = 864 Useful for teacher led demonstration.
II.
+ 368
496
14
150
700
864
III.
368
496
864
11
NB. When adding decimals, line up decimal points.
7) Addition of several numbers with different numbers of digits, understanding place
value.
37
37 + 129
129
166
1
Year 4 to consolidate use in money and in length.
8) Year 5 - Addition of whole numbers with more than 4 digits & decimals with 2 places.
Addition of money & measures (including problems where you have to
convert from one unit to another e.g. 4.3kg + 125g.)
9) Year 6 addition of whole numbers & decimals with any number of digits. e.g.
3
42
6432
786
3
4681
11944
401.
26.
0.
428.
1 1 2 1
1
2
8
7
7
0
5
1
6
When adding decimals with different numbers
of decimal places, children should be
encouraged to make them the same, by
adding zeros, understanding that 2 tenths is
the same as 20 hundredths, therefore 0.2 is
the same as 0.20
Stages in Teaching Subtraction
- 40
-3
1) Use of a number line e.g. 76 - 43 _________________________________
33
73
76
2) Horizontal written expansion could be used during teacher explanation, but not by
pupils. E.g.
7 6 - 4 3 = (70 - 40) + (6 - 3)
= 30 + 3
= 33
3) Vertical written expansion; without decomposition.
e.g. 7 6 - 4 3
76=70
6
Use of base 10 apparatus
- 43=40
3
important here.
30
3 = 33
Remember to emphasise that the bottom number is being subtracted from the top.
NOT the lesser number from the greater.
4) Vertical written expansion, with decomposition.
20
1
37=30
7
-1 9 = 1 0
9
10
8 =18
Use of base 10 apparatus continues to be very important. Usual target for most Y3
children.
5a) Vertically subtracting the least significant figure first.
7 6 Use only for teacher demonstration if necessary.
-43
4
3
30
33
5b) Vertically with decomposition.
81
- 9 3 Use only for teacher demonstration if necessary.
47
6
40
46
6) Compact vertical form.
3 1
48
4 8 This should be the target for most Y3 pupils
-24
- 3 9 using up to 3 digits.
24
9
7) Year 4 :- use of larger numbers with up to 4 digits and decimals with one decimal
place.
30
1
342=300
40
2
126=100
20
6
=200
10
6 = 216
31
342
-1 2 6
216
N.B. When subtracting decimals line up the decimal points.
1
400
563=500
-278=200
200
Then :4
15
5
- 2
2
6
7
8
50
60
70
80
1
3
8
5 = 285
1
3
8
5
By the end of year 4, children should be using the written method confidently and
with understanding.
5
They will also be subtracting numbers with a different numbers of digits,
understanding the place value, and decimals with one decimal place.
8) Year 5 :- Extend to subtracting whole numbers with more than 4 digits and
decimals with 2 decimal places.
2 13 1
3.42
- 1.76
1.66
9) Year 6 :- Extend the decomposition method and use it to subtract whole numbers
and decimals with any number of digits.
3 1 6 11 1
When subtracting decimals with different
417.20
numbers of decimal places, children should
- 34.71
be taught and encouraged to make them
382.49
the same through the knowledge that
2 tenths is the same as 20 hundredths,
therefore 0.2 is the same as 0.20
10) Subtract amounts of money and measures, including those where you have to
initially convert one unit to another.
Generally children will be ready to meet standard written methods for division and
multiplication when they have mastered the following: Knowledge of 2, 3, 4, 5, 6, 7, 8, 9 and 10 times tables and the corresponding
division facts.
 Know the results of multiplying by 0 or 1.
 Understand place value.
 Understand value of 0 as a place holder.
 Be able to multiply 2 and 3 digit numbers by 10 and 100 mentally.
 Understand cumulative and associative laws for multiplication.
E.g. 4 x 3 = 3 x 4
12 ÷ 3 = 4
12 ÷ 4 = 3
 Able to double and halve 2 digit numbers mentally.
 Able to explain their mental strategies orally.
Stages in Teaching Multiplication.
6
1) Single digit x single digit in array e.g. 2 x 3 is 2 rows of 3.
2
3
2) Repeated addition e.g. 2 x 3 = 3 + 3 = 6 Using Cuisenaire rods.
3) Use of jumping along number line. E.g. 5 x 3 = 15
+3
0
+3
3
+3
6
+3
9
+3
12
15
4) Grid layout, expanded working.
e.g. 38 x 7 = (30 x 7) + (8 x 7)
7 Do not labour this, just use it to illustrate multiplying the tens and the
ones separately.
30 210
8
56
266
5) Vertical format.
e.g. 3 8
x 7
5 6 ( 8 x 7)
2 1 0 (30 x 7)
266
5
Teacher demonstration of this may be all that is required, in order to explain move to
compact format.
6) Compact vertical format.
e.g. 3 8
This is target stage for able Y3 pupils.
7
X 7
266
5
7) Year 4 consolidate, multiplying 2 digit and 3 digit numbers by a one digit number.
e.g. 346 x 8
346
X
8
2768
3 4
8) Year 5.
Multiplying numbers up to 4 digits by one or two digit numbers.
e.g. 143 x 26
Demonstrate that this is the same as (143 x 20) + ( 143 x 6)
Then carry out 2 separate compact vertical forms.
143
143
Then 2 8 6 0
x20
x 6
858+
2860
858
3718
2 1
NB. When multiplying by a multiple of 10, explain that x 20 = x10 x 2. Therefore
x 10 first, which results in moving all digits one place to the left, so inserting a zero
into the ones column as a place holder. Then multiplying by 2.
E.g.
143
This is an important concept and should be extended into
X 10
multiplying by all multiples of 10 eg 30, 40 etc.
1430
X2
2860
Children should be able to use this method to solve problems involving money and
measures.
9) Year 6. Extend to multiply numbers with up to 2 decimal places by whole numbers.
8
4.92
X
3
14.76
2
Children should be able to use this method to solve problems and multiplying
numbers, including those with decimals, in the context of money or measures.
E.g. to calculate the cost of 7 items at £8.63 each, or the total length of 6 pieces of
ribbon of 2.28m each.
Stages in Teaching Division
There are two ways of approaching division, Cover both approaches.
E.g. “Shares” - as in share 15 sweets between 3 children.
(Use of base 10 apparatus)
“Goes into’’
- As in how many times does 6 go into 18, or how many 6s in 18.
(Use of Cuisenaire rods)
1) Year 3 – Division of 2 digit numbers by a single digit using Cuisenaire rods,
multiplication squares and knowledge of tables.
E.g. 32 ÷ 8 = 4
E.g. 46 ÷ 7 = 6 r 4
Children need to be able to make decisions about what to do with remainders and
round up or down accordingly.
E.g. In the calculation 13 ÷ 4 the answer is 3 remainder1, but in problems,
whether the answer should be rounded up to 4 or rounded down depends on the
context as below.
a) I have £13. Books are £4 each. How many can I buy?
Answer 3 (The remaining £1 is not enough to buy another book.)
b) Apples are packed in to boxes of 4. There are 13 apples. How many boxes are
needed?
Answer 4 (The remaining 1 apple still needs to be placed in a box.)
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2) Year 4 Division of numbers up to 3 digits by a 1 digit number, using base 10 then
without apparatus.
1 4
E.g
1 4
E.g.
1
rem. 1 Known as " Bus
Stop" Method.
1
4 5 6
4 5 7
N.B. Introduce decimal values with money and measurement.
E.g.
£1 .5 0
1
2 3 .0 0
3) By year 5, division of numbers up to 4 digits by one digit numbers. Remainder can
be expressed as a fraction or a decimal. See Year 3 for remainders in written
problems.
4) In year 6, use continuous subtraction or "Chunking" when dividing numbers up to
4 digits by a 2 digit number. (Continue "Bus Stop" method when dividing by a one
digit number)
E.g.
1
12 1 4
12
2
2
0
2
4
0
4
4
0
3
E.g.
3
1
0 3
3 0 (13 x 10)
7 3
3 0 (13 x 10)
4 3
3 0 (13 x 10)
1 3
1 3 (13 x 1)
0 0
Continuous subtraction with remainders. Use all 3 ways of expressing remainders,
depending on the question.
E.g. a)
12 x 10
12 x 2
1
13 4
1
2
1
1
1
1 8 rem 9
b)
10
3 8 1/12
13 2 4 3
1 3 0
0
12 4 5 7
3 6 0
9 7
4 8
4 9
4 8
1
(13 x 10)
101
1 1
2
8
6
2
1
3
6
7
5
2
3
9
(13 x 2)
(13 x 5)
(12 x 30)
1 remainder
12 divisor
(12 x 4)
(12 x 4)
(13 x 1)
c) To show remainders as a decimal relies upon the children's knowledge of decimal
fraction equivalents. E.g. 3 3/5 = 3.6
See year 3 re. Remainders in written problems.
December 2015
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