Download sample policy

Progression in Written Methods for the Four Operations
This policy has been written in consultation with all teaching staff.
Children will be taught mental methods from Foundation Stage. They will also learn to use
jottings to support them with calculations. By the time children reach late Year 3 the numbers
they are working with are less manageable mentally or with jottings so they need written
methods.
The policy outlines the mental methods children need to be secure in to be successful with our
chosen written methods, as well as the written methods themselves.
When children start to learn a written method for any of the four operations they are still learning
mental methods. When children meet a calculation they should be encouraged to ask themselves:
1) Can I do this mentally?
2) Can I do this mentally with jottings?
3) Do I need a written calculation? (from late Year 3 for + and -, from early Year 4 for x and )
4) Do I need a calculator? (from Year 5)
1
Mental methods children are taught which will enable them to effectively use empty
number lines and complementary addition for subtraction
Reception
Year 1
Year 2
Year 3
Year 4
Recite the number names in order, continuing the count forward or backwards
from a given number.
Begin to relate subtraction to ‘taking away’.
Remove a smaller number from a larger number and find how many are left
by counting back.
Begin to find out how many have been removed from a larger group of objects
by counting up from a number.
Work out by counting how many more are needed to make a larger number.
Count on or back in ones from any small number, and in tens from and back to
zero.
Begin to know what each digit in a two-digit number represents. Partition a
‘teens’ number and begin to partition larger two-digit numbers.
Within the range 0 to 30, say the number that is 1 or 10 more or less than any
given number.
Know by heart all pairs of numbers with a total of 10.
Begin to know addition facts for all pairs of numbers with a total of at least
10, and the corresponding subtraction facts.
Use known number facts and place value to add or subtract a pair of numbers
mentally within the range 0 to at least 10, then 0 to at least 20.
Count on or back in ones or tens, starting from any two-digit number.
Know what each digit in a two-digit-number represents, and partition twodigit numbers into a multiple of tens and ones.
Say the number that is 1 or 10 more or less than any given two-digit number.
Know by heart all addition and subtraction facts for each number to at least
10, all pairs of numbers with a total of 20, all pairs of multiples with a total of
100.
Find a small difference by counting up from the smaller to the larger number.
Add/subtract 9/11 by adding/subtracting 10 and adjusting by 1.
Begin to add/subtract 19/21.
Use known number facts and place value to add/subtract mentally.
Count on or back in tens or hundreds, starting from any two- or three-digit
number.
Know what each digit represents, and partition three-digit numbers into a
multiple of 100, a multiple of ten and ones.
Say the number that is 1, 10 or 100 more or less than any two- or three-digit
number.
Know by heart all addition and subtraction facts for each number to 20 and all
pairs of multiples of 100 with a total of 1000.
Derive quickly all pairs of multiples of 5 with a total of 100.
Find a small difference by counting up from the smaller to the larger number.
Add and subtract mentally a near multiple of 10 to or from a two-digit
number, by adding or subtracting a multiple of ten then adjusting.
Partition numbers into thousands, hundreds, tens and ones.
Add/subtract 1, 10, 100 or 1000 to/from any integer and count on or back in
tens, hundreds or thousands from any whole number up to 10 000.
Consolidate knowing by heart addition and subtraction facts for all numbers to
20.
Derive quickly all number pairs that total 100, all pairs of multiples of 50 with
a total of 1000.
Find a small difference by counting up.
Count on or back in repeated steps of 1, 10 or 100.
2
Year 5
Year 6
Add or subtract the nearest multiple of 10 then adjust.
Use known number facts and place value to add or subtract mentally.
Know what each digit represents in a number with up to two-decimal places.
Derive quickly decimals that total 1, all two-digit pairs that total 100, all pairs
of multiples of 50 with a total of 1000.
Find differences by counting up through next multiple of 10, 100 or 1000.
Add/subtract the nearest multiple of 10 or 100 then adjust.
Use known number facts or place value for mental addition and subtraction.
Know what each digit represents in a number with up to three decimal places.
Consolidate finding a difference by counting on.
Add/subtract the nearest multiple of 10, 100 or 1000, then adjust.
Use known number facts and place value to consolidate mental
addition/subtraction.
All the above objectives help consolidate understanding of subtraction questions using empty
number lines and complementary addition.
“If an emphasis on mental calculation is expected to lead to the development of written
algorithms based on these strategies then it is important to ask why we need to introduce
subtraction by decomposition. This appears to represent a confusion in the aims, and to my
knowledge, there are no references in the research or professional literature, on children’s
idiosyncratic mental algorithms, to any children having invented or discovered the decomposition
algorithm for themselves.”
Ian Thompson (Issues for Classroom Practice in England)
3
Mental methods children are taught which will enable them to effectively use the grid
method of multiplication
Reception
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Count in tens.
Count in twos.
Count on and back in tens from and back to zero.
Count on in twos from zero.
Count in steps of five from zero.
Begin to count in steps of three from zero.
Begin to know what each digit in a two-digit number represents. Partition a ‘teens’
number and begin to partition larger two-digit numbers into a multiple of ten and
ones.
Count in hundreds from and back to zero.
Count on in twos from and back to zero.
Count in steps of 3, 4 or 5 to at least 30, from and back to zero.
Know what each digit in a two-digit number represents, and partition two-digit
numbers into a multiple of ten and ones.
Understand the operation of multiplication as repeated addition or as describing an
array.
Know by heart multiplication facts for the 2 and 10 times-tables.
Use known number facts and place value to carry out mentally simple
multiplications.
Derive quickly doubles of all multiples of numbers to at least 15 and multiples of 5
to 50.
Count on in steps of 3, 4 or 5 from any small number to at least 50 and then back
again.
Know what each digit represents, and partition three-digit numbers into a multiple of
100, a multiple of ten and ones.
Know by heart multiplication facts for the 2, 5 and 10 times-tables.
Begin to know the 3 and 4 times-tables.
To multiply by 10/100, shift the digits one/two places to the left.
Use known number facts and place value to carry out mentally simple
multiplications.
Partition numbers into thousands, hundreds, tens and ones.
Multiply and divide any integer up to 1000 by 10 and understand the effect. Begin
to multiply by 100.
Know by heart multiplication facts for 2, 3, 4, 5 and 10 times-tables.
Partition (e.g. 23 x 4 = (20 x 4) + (3 x 4)).
Use known facts and place value to multiply integers, including by 10 and then 100.
Know what each digit represents in a number with up to two decimal places.
Multiply and divide any positive integer up to 10 000 by 10 or 100 and understand
the effect.
Know by heart all multiplication facts up to 10 x 10.
Partition (e.g. 47 x 6 = (40 x 6) + (7 x 6)).
Use known facts and place value to multiply mentally.
Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and
explain the effect.
Consolidate knowing by heart multiplication facts up to 10 x 10
Partition (e.g. 87 x 6 = (80 x 6) + (7 x 6)).
Use known number facts and place value to consolidate mental multiplication.
4
Mental methods children will need in order to divide using ‘chunking’
Reception
Year 1
Year 2
Year 3
Year 4
Count in tens.
Count in twos.
Count on and back in tens from and back to zero.
Count on in twos from zero.
Count in steps of five from zero.
Begin to count in steps of three from zero.
Begin to know what each digit in a two-digit number represents. Partition a ‘teens’
number and begin to partition larger two-digit numbers into a multiple of ten and ones.
Count in hundreds from and back to zero.
Count on in twos from and back to zero.
Count in steps of 3, 4 or 5 to at least 30, from and back to zero.
Know what each digit in a two-digit number represents, and partition two-digit numbers
into a multiple of ten and ones.
Understand the operation of division as grouping (repeated subtraction) or as sharing
equally.
Interpret 8  2 as ‘how many 2s make 8?’
Know by heart multiplication and associated division facts for the 2 and 10 times-tables
and derive quickly the corresponding division facts.
Use known number facts and place value to carry out mentally, simple divisions.
Derive quickly halves of even numbers to 20 and begin to halve of multiples of 10 up to
100.
Count on in steps of 3, 4 or 5 from any small number to at least 50 and then back again.
Know what each digit represents, and partition three-digit numbers into a multiple of
100, a multiple of ten and ones.
Know by heart multiplication and associated division facts for the 2, 5 and 10 timestables.
Know that dividing a whole number by 1 leaves the number unchanged.
Begin to know the 3 and 4 times-tables.
Understand that 16  2 does not equal 2  16.
Understand that division reverses multiplication and solve division calculations by using
multiplication strategies.
Understand the idea of a remainder.
Make sensible decisions about rounding up or down after a division in the context of a
problem.
To divide by 10, shift the digits one place to the right.
Use known number facts and place value to carry out mentally, simple divisions.
Say a division statement corresponding to a given multiplication statement.
Begin to relate division and fractions.
Partition numbers into thousands, hundreds, tens and ones.
Multiply and divide any integer up to 1000 by 10 and understand the effect. Begin to
divide by 100.
Know by heart multiplication and associated division facts for 2, 3, 4, 5 and 10 timestables.
Use known facts and place value to multiply and divide integers, including by 10 and
then 100.
Use known facts and place value to add or subtract a pair of numbers mentally.
Use doubling and halving, and factors.
5
Year 5
Year 6
Relate division and fractions.
Know what each digit represents in a number with up to two decimal places.
Multiply and divide any positive integer up to 10 000 by 10 or 100 and understand the
effect.
Know by heart all multiplication and associated division facts up to 10 x 10.
Use known facts and place value to add or subtract a pair of numbers mentally.
Use known facts and place value to multiply and divide mentally.
Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain
the effect.
Consolidate knowing by heart multiplication and associated division facts up to 10 x 10
Partition (e.g. 87 x 6 = (80 x 6) + (7 x 6)).
Use known number facts and place value to consolidate mental division.
6
Progression in Addition
The right hand side refers to the language used to explain the calculation, children will not write
the explanation.
1. T U add T U adding most significant digits first
+
6
2
8
1
9
7
4
0
1
1
60 + 20
7+4
80 + 11
2. T U add T U adding least significant digits first
+
1
1
7
4
1
1
2
5
8
3
0
3
5+8
70 + 40
110 + 13
3. H T U add T U adding least significant digit first, preparing for carrying
3
+
1
3
4
5
7
1
2
0
3
8
3
1
0
0
1
8+3
50 + 70
300 + 0
300 + 120 + 11
4. H T U add T U carrying below the line
6
+
6
2
4
7
5
8
3
1
5. H T U add H T U using carrying
+
1
5
4
0
8
7
6
1
1
7
5
2
Children will then learn to add Th H T U + H T U then Th H T U + Th H T U and decimals. If at
any stage, children get ‘stuck’ they should return to the expanded method.
7
Progression in Subtraction
Whilst the right hand side refers to the language used to explain the calculation, children will be
learning to write the explanation, leaving the number line behind as appropriate.
1. Counting up from the smaller to the larger number (complementary addition)
84 – 56
-
-
84
56
4 to make 60
20 to make 80
4 to make 84
28
783
356
4
40
300
83
427
to make 360
to make 400
to make 700
to make 783
Children can continue to use number lines rather than move to a vertical format.
2. H T U – T U Counting Up
-
754
86
4
10
600
50
4
668
90
100
700
750
754
3. H T U - H T U Counting Up
-
754
286
14
400
54
668
300
700
754
8
4. Th H T U – H T U Counting Up
5. Th H T U – Th H T U Counting Up
-
6467
2684
16
300
3467
3783
2700
3000
6467
9
Progression in Multiplication
The right hand side refers to the language used to explain the calculation, children will not write
the explanation.
1. T U x U Grid Method
x
8
20
160
3
24
= 184
40
360
6
54
2. H T U x U
x
9
300
2700
= 3114
3. T U x T U
x
30
8
70
2100
560
2
60
16
2160
+576
2732
4. Th H T U x U
x
8
4000
32 000
300
2400
40
320
6
48
= 34 678
5. H T U x T U
x
20
4
300
6000
1200
70
1400
280
2
40
8
7440
+1488
8928
5. H T U x U . t
x
2
0.4
300
600
120
70
140
28
2
4
0.8
744
+148.8
892.8
10
Progression in Division
1. T U  U
96  6 =
6 96
- 60
36
- 36
0
10 x 6
6x6
= 16
2. H T U  U
196  6 =
6 196
- 180
16
- 12
4
30 x 6
2x6
= 32 r. 4
3. H T U  T U
972  36 =
36 972
- 720
252
- 252
0
20 x 36
7 x 36
= 27
11