x. . ÷. +. Birchwood Primary School. Progression in Calculations. Introduction.
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Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly.These skills lead on to more formal written methods of calculation.Strategies for calculation need to be supported by familiar models, images and practical activities to reinforce understanding. When teaching a new strategy it is important to start with numbers the child can easily manipulate so that they can understand the concept. The transition between levels should not be hurried. Children should not be moved onto the next method until they are secure in their current method and are able to explain it. Previous levels should be revisited to consolidate understanding when introducing a new strategy.A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.
8
+
+
1
2
3
4
5
6
7
8
9
10
One more than three is four
1, 2, 3, 4, 5, 6 … there are 6 teddies
P8 or ELG’s
Recognise numbers 0 to 10
Level 1c
Count reliably up to 10 everyday objects
Level 1c
Find one more than a number
Level 1c
Begin to relate addition to combining two groups of objects
makes 5
and
Level 1b
Count in ones and tens
Level 1b
Count along a number line to add numbers together
3 + 2 = 5
Level 1b
6 + 4 = 10
Begin to use the + and = signs to record mental calculations in a number sentence
=
+
=
1
1
2
2
3
3
Level 1a
Know that addition can be done in any order
Level 1a
3 +5
Put the biggest number first and count on
Level 2c
Know doubles of numbers
+
5 5 = 10
Level 2b
8 + 7 = 15
Add two singledigit numbers that bridge 10
1
5
9
6
2
4
8
3
7
Level 2b
Know by heart all pairs of numbers with a total of 10
Level 2a
Begin to partition numbers in order to add
15
11
15
9
6
12
14
8
13
7
15
16
17
18
25
26
27
28
15
25
28
Level 2a
15 + 1 = 16
Know which digit changes when adding 1s or 10s to any number
15
16
15 + 10 = 25
25
15 + 20 = 35
15
25
35
Level 2a
Adding two twodigit numbers (without bridging)
Counting in tens and ones
Partitioning and recombining
15 + 13 = 28
Level 3c
Know by heart all pairs of numbers with a total of 20
6 and how many more make 20?
8
30
6
Level 3c
Add near multiples of 10 to a 2 digit number
25 + 21 = 46
Level 3c
Adding two twodigit numbers (bridging through tens boundary)
Using a number line
OR
Using place value cards and Dienes to partition numbers and recombine
48 + 36 = 84
40 + 30 + 8 + 6
40 + 30 = 70
8 + 6 = 14
70 + 14 = 84
T
U
Expanded method
It is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.
48 + 36
48
+36
T U
40 + 8
30 + 6
80 + 4
10
This method should also be used for adding HTU to HTU
Level 3a
4 8
+ 3 6
8 4
Standard written method
The previous stages reinforce what happens to the numbers when they are added together using more formal written methods.
1
This method should also be used for adding HTU to HTU
528 + 288
500 + 200 = 700
20 + 80 = 100
8 + 8 = 16
= 816
They must also add HTU mentally, starting wit the most significant digit first.
Level 4b +
£4.85
+ £3.38
£8.23
1 1
Add must add decimals to 2 decimal places
8


9
8
7
,
,
,
...
P8 or ELG’s
Ten green bottles hanging on the wall …
Five fat sausages frying in a pan …
Begin to count backwards in familiar contexts such as number rhymes or stories
Level 1c
Continue the count back in ones from any given number
Level 1c
Three teddies take away two teddies leaves one teddy
Begin to relate subtraction to ‘ taking away ’
Level 1c
Find one less than a number up to 10
Level 1b
Count back in tens
Level 1b
6  4 = 2
Begin to use the – and = signs to record a number sentence using numbers up to 10 whilst still using practical resources
The difference between 8 and 5 is 3
Level 1b
1
1
2
2
3
3
Compare 2 sets to find numerical difference
Level 1a
Maria had six sweets and she ate four. How many did she have left?
Understand the operation of subtraction and use related vocab. Subtract numbers when solving problems involving up to 10 objects in a range of contexts
6  4 = 2
Level 2c
13 – 1 =
21 – 10 =
Within the range 1 30 say 1 or 10 less than any number
Level 2b
7 – 2 = 5
Know by heart subtraction facts for numbers up to 10
15  7 = 8
Subtract single digit numbers often bridging through 10
Level 2b
Subtract 1 from a twodigit number
45  1
Level 2a
Begin to find the difference by counting up from the smallest number
Level 2a
Subtract 10 from a twodigit number
20
30
25
45
20
23
33
43
43 –
43 – 20 = 23
23 – 3 = 20
20
3
 3
 10
 10
Level 2a
Subtract multiples of 10 from any number
Level 2a
Begin to partition numbers in order to take away
Partition the number to be subtracted (no exchanging)
Level 3c
Decide whether to count on or count back
74  27 = 47
Now where’s the answer?
U
43  27 = 16

2
7
to subtract 7 units we need to exchange a ten for ten units
40 + 3
 20 + 7
10 + 6
30
10 +
Level 3c
Subtract mentally a near multiple of 10 from a 2 digit number
56 – 21 = 35
Level 3c
Know by heart subtraction facts for numbers up to 10
Level 3b
Expanded method
It is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.
 2 7
1 6
3
1
Level 3a
Standard written method
The previous stages reinforce what happens to numbers when they are subtracted using more formal written methods. It is important that the children have a good understanding of place value and partitioning.
Level 4b
Subtract decimals to 2 places
²3 .¹79ml
 2. 86ml
______
0 .93ml
https://www.ncetm.org.uk/resources/40532
8
Progression in Teaching Multiplication
Mental Skills
Recognise the size and position of numbers
Count on in different steps 2s, 5s, 10s
Double numbers to 10
Recognise multiplication as repeated addition
Quick recall of multiplication facts
Use known facts to derive associated facts
Multiplying by 10, 100, 1000 and understanding the effect
Multiplying by multiples of 10
Models, Images and apparatus
Place value apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Counting stick
Models and Images charts
ITPs – Multiplication grid, Number Dials, Multiplication Facts
Vocabulary
Lots of
Groups of
Times
Multiply
Multiplication
Multiple
Product
Once, twice, three times
Array, row, column
Double
Repeated addition
x
x
2
4
6
8
0
5
10
15
20
25
Level 1b
Count in tens from zero
0
20
30
40
50
Level 2c
Count in twos from zero
Level 2c
Count in fives from zero
Level 2b
Know doubles and corresponding halves to 20
+
+
+
2 4
x
4 2
x
Level 2a
2 + 2 + 2 + 2 = 8
4 x 2 = 10
2 multiplied by 4
4 lots of 2
Understand multiplication as repeated addition
Level 2a
Understand multiplication as an array
Level 2a
Understand how to represent arrays on a number line
x 5
Level 2a
2 x 5 = 10
6 x 5 = 30
Tables spider
3 x 5 = 15
4 x 5 = 20
10 x 5 = 50
8 x 5 = 40
Know by heart facts for the 2,5 & 10 multiplication tables
5 x 5 = 25
Use known facts to work out new ones
Use known facts and place value to carry out mentally simple x calculations
Level 3c
Know by heart facts for the 2,3,4,5 & 10 multiplication tables
10
3
3
4
4
40
12
10
3
40
12
4
10
10
3
4
40
40
12
20 ( 2 x 10 )
3
4
80
12
Level 3c
Use place value apparatus to support the multiplication of U x TU
4 x 13
Level 3c
Use place value apparatus to support the multiplication of U x TU alongside the grid method
4 x 13
40 + 12 = 52
Level 3b
Use place value apparatus to represent the multiplication of U x TU alongside the grid method
4 x 23
80 + 12 = 92
3
10
300
30
= 330 +
4
120
12
= 132
462
56 × 27 1120 (56 × 20) 392 (56 × 7)1512
1
Level 3a +
14 x 33
Multiplying TU x TU
300
120
30
+ 12
462
This may be as far as many children progress in their use of multiplication.
Level 4c
Standard written method
Level 4b
Use factors to multiply
Understand that …
24 x 20 = 24 x 2 x 10
24 x 50 = 24 x 5 x 10
https://www.ncetm.org.uk/resources/40530
8
Progression in Teaching Division
Mental Skills
Recognise the size and position of numbers
Count back in different steps 2s, 5s, 10s
Halve numbers to 20
Recognise division as repeated subtraction
Quick recall of division facts
Use known facts to derive associated facts
Divide by 10, 100, 1000 and understanding the effect
Divide by multiples of 10
Models, Images and apparatus
Counting apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
ITPs – Multiplication, remainders grid, Number Dials, Grouping,
Vocabulary
Lots of
Groups of
Share
Shared between
Group
Divide
Divide into
Division
Divided by
Remainder
Factor
Quotient
Divisible
÷
÷
10
20
30
?
2
4
6
8
Level 1c
Give every bear a sweet – how many sweets do we need?
Solve problems by sharing objects in a practical or role play context
Level 1b
Count back in tens
Give each bear 2 sweets – how many sweets do we need?
Level 1a
Solving practical problems by sharing into equal groups
Level 2c
Count back in twos
Level 2c
Count back in fives
0
5
10
15
Half of 6 is 3
½ of 6 = 3
Know halves to 20
Level 2a
Understand division as sharing
1 2 3 4 5
6 7 8 9 10 etc….
Level 2a
Understand division as grouping
1,2,3,4,5 6,7,8,9,10 etc
Level 2a
12 divided into groups of 3 gives 4 groups
12 3 = 4
12 divided into groups of 4 gives 3 groups
12 4 = 3
Reinforce division as grouping through the use of arrays
3
6
9
12
15
18
Level 3c
18 divided into groups of 3
18 3 = 6
Represent ‘groups’ for division on a number line using apparatus alongside the line to include remainders.
18 3 = 6
3
3
3
3
3
3
0
18
3
6
9
12
15
18 6 = 3
Level 3b
126 ÷ 21 = 6
21
21
21
21
21
21
84
105
126
0
21
42
63
180 ÷ 60 = 30
Level 3a
126 ÷ 21 = 6
(2 x21)
(2 x21)
(2 x21)
0
42
84
126
https://www.ncetm.org.uk/resources/43589
This must not be introduced to the children before Year 5
18
(1 x 3)
15
Understand division as repeated subtraction using a vertical line and apparatus to make the links
(1 x 3)
12
(1 x 3)
9
(1 x 3)
6
(1 x 3)
3
(1 x 3)
0
What facts do I know about the 7 timestable?
Year 5 and 6 only once confident with previous method
Children need to see that as the numbers get larger, large chunk subtraction is the more efficient method. Multiples of the divisor (large chunks) are taken away. Multiplication facts are needed to see the size of the ‘chunk’.
Fact Box
1 x 7 = 7
2 x 7 = 14
5 x 7 = 35
10 x 7 = 70
20 X 7 = 140
50 x 7 = 350
100 x 7 = 700
518 ÷ 7 = 74
518
 350 ( 50 x 7 )
168
 140 ( 20 x 7 )
28
 28 ( 4 x 7 )
0
100 ÷ 7 = 14 r 2
100
 70 ( 10 x 7 )
30
 28 ( 4 x 7 )
2
Level 4a
Year 6 Top set only
560 ÷ 24
2 3 r 8
2 4 5 6 0
4 8 0
8 0
7 2
8
Standard written method
Links directly to large chunk subtraction