Transcript Progression in Calculations Written methods of

Aston by Sutton
Primary School
÷
1
Introduction
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.
2
Progression in Teaching Addition
Mental Skills
• Recognise the size and position of numbers
• Count on in ones and tens
• Know number bonds to 10 and 20
• Add multiples of 10 to any number
• Partition and recombine numbers
• Bridge through 10
Models, Images and apparatus
• Place value apparatus - diennes
• Arrow cards
• Number tracks
40
8
• Numbered number lines
• Marked but unnumbered number lines
• Empty number lines
• Hundred square
• Counting stick
• Bead string
• Models and Images charts
• ITPs – Number Facts, Ordering Numbers, Number Grid, Counting on and back in ones and tens
Key Vocabulary
• add
• addition
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase
P8 or ELG’s
Recognise numbers 0 to 10
Level 1c
1, 2, 3, 4, 5, 6
… there are 6
teddies
Count reliably up to 10 everyday objects
Level 1c
Find one more than a number
One more than
three is four
Level 1c
Begin to relate addition to
combining two groups of objects
Level 1b
Count in ones and tens
Level 1b
3+2=5
Count along a number line to
add numbers together
Level 1b
Begin to use the + and = signs to record
mental calculations in a number sentence
6 + 4 = 10
4
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
Level 2b
8 + 7 = 15
+2
8
Add two single-digit
numbers that bridge 10
+5
10
15
5
Level 2b
Know by heart all pairs of numbers
with a total of 10
3 7
Level 2a
Begin to partition numbers
in order to add
6
Level 2a
15 + 1 = 16
Know which digit
changes when
adding 1s or 10s
to any number
15
16
15 + 10 = 25
25
15
15 + 20 = 35
15
Level 2a
25
35
15
16
17
18
Adding two two-digit numbers
(without bridging)
25
26
27
28
Counting in tens and ones
Partitioning and recombining
15
25
28
15 + 13 = 28
Level 3c
Know by heart all pairs of numbers
with a total of 20
13 7
6 and how many
more make 20?
7
Level 3c
Add near multiples of 10 to a 2
digit number
25 + 21 = 46
+30
Level 3c
+2
48
+2
Adding two two-digit numbers
(bridging through tens boundary)
Using a number line
OR
Using place value cards and Dienes
to partition numbers and recombine
48 + 36 = 84
78
48
+4
80
+34
84
50
40
84
8
30
6
40 + 30 + 8 + 6
40 + 30 = 70
8 + 6 = 14
70 + 14 = 84
8
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.
T
U
48 + 36
48
T
+ 36
U
40 + 8
30 + 6
This method should also be used for
adding HTU to HTU
80 + 4
10
Level 3a
48
+ 36
84
1
Standard written method
The previous stages reinforce what
happens to the numbers when they
are added together using more
formal written methods.
This method should also be used for
adding HTU to HTU
9
Level 4c
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
Add must add decimals to 2 decimal
places
£8.23
1
1
10
Progression in Teaching Subtraction
Mental Skills
•
•
•
•
•
•
Recognise the size and position of numbers
Count back in ones and tens
Know number facts for all numbers to 20
Subtract multiples of 10 from any number
Partition and recombine numbers (only partition the number to be subtracted)
Bridge through 10
Models, Images and apparatus
Place value apparatus - diennes
Arrow cards
40
8
Number tracks
Numbered number lines
Marked but unnumbered lines
Hundred square
Empty number lines.
Counting stick
Bead strings
Models and Images Charts
ITPs – Number Facts, Counting on and back in ones and tens, Difference
Key Vocabulary
Subtract
Take away
Minus
Count back
Less
Fewer
Difference between
11
P8 or ELG’s
Begin to count backwards in
familiar contexts such as
number rhymes or stories
Level 1c
Level 1c
Begin to relate subtraction
to ‘ taking away ’
Five fat sausages
frying in a pan …
Ten green bottles
hanging on the wall
…
Continue the count back in
ones from any given number
Three teddies take away
two teddies leaves one
teddy
Level 1c
Find one less than
a number up to 10
Level 1b
Count back in tens
Level 1b
Begin to use the – and = signs
to record a number sentence
using numbers up to 10 whilst
still using practical resources
6-4=2
12
Level 1b
1
2
3
The
difference
between 8
and 5 is 3
Compare 2 sets to find
numerical difference
Level 1a
Understand the operation of
subtraction and use related
vocab. Subtract numbers when
solving problems involving up to
10 objects in a range of
contexts
Level 2c
21 – 10 =
Maria had six sweets and
she ate four. How many
did she have left?
6-4=2
13 – 1 =
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
13
Level 2b
15 - 7 = 8
Subtract single digit
numbers often bridging
through 10
Level 2b
Subtract 1 from a
two-digit 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
14
Level 2a
25
45
Subtract multiples of
10 from any number
Level 2a
Begin to partition numbers in
order to take away
-3
20
43 –
43 – 23
- 10
23
3
43 – 20 = 23
- 10
33
20
43
23 – 3 = 20
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?
15
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
43 - 27 = 16
Expanded method
to subtract 7 units
we need to exchange
a ten for ten units
T
- 2
U
7
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.
40
+
- 20
+
7
10
+
6
30
10 +
3
16
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.
3
4 13
- 2 7
1 6
Level 4b
Subtract decimals to 2 places
² 3 .¹79ml
- 2. 86ml
______
0 .93ml
17
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
40
8
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
18
Level 1b
Count in tens
from zero
0
20
30
40
50
Level 2c
Count in twos
from zero
0
2
4
6
8
Level 2c
Count in fives
from zero
0
Level 2b
5
10
15
20
25
Know doubles and
corresponding halves to 20
19
Level 2a
2+2+2+2=8
Understand multiplication as
repeated addition
4 x 2 = 10
2 multiplied by 4
4 lots of 2
Level 2a
Understand multiplication
as an array
Level 2a
Understand how to
represent arrays
on a number line
x5
Level 2a
Tables
spider
2 x 5 = 10
6 x 5 = 30
3 x 5 = 15
4 x 5 = 20
10 x 5 = 50
Know by heart facts for the 2,5 & 10
multiplication tables
8 x 5 = 40
5 x 5 = 25
20
Level 3c
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
21
Level 3c
10
Use place value apparatus to support
the multiplication of U x TU
3
4
4 x 13
Level 3c
4
4
10
3
40
12
10
3
40
12
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
10
4
10
40
20
4 x 23
4
3
40
12
( 2 x 10 )
3
80
12
80 + 12 = 92
22
Level 3a +
14 x 33
Multiplying TU x TU
30
3
10
300
30
= 330 +
4
120
12
= 132
+ 12
462
462
This may be as far as many
children progress in their use
of multiplication.
300
120
30
Level 4c
56
× 27
1120
392
1512
(56 × 20)
(56 × 7)
Standard written method
1
Level 4b
Use factors to multiply
Understand that …
24
x 20 = 24 x 2 x 10
24
x 50 = 24 x 5 x 10
23
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
40
8
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
÷
24
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
0
10
20
Level 1a
Give each
bear 2
sweets – how
many sweets
do we need?
Solving practical
problems by sharing into
equal groups
Level 2c
?
30
Count back in twos
2
4
6
8
Level 2c
Count back in fives
0
5
10
15
25
Level 2b
Know halves to
20
Half of 6 is 3
½ of 6 = 3
Level 2a
Understand division
as sharing
1
2
3
4
6
7
8
9
5
10 etc….
Level 2a
Understand division
as grouping
1,2,3,4,5
Level 2a
6,7,8,9,10 etc
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
26
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.
0
18  3 = 6
-3
3
6
-3
9
-3
-3
12
-3
15
18
-3
0
18
3
Level 3b
6
-21
21
Level 3a
-21
42
-21
-21
-21
84
105
180 ÷ 60 = 30
63
126
126 ÷ 21 = 6
(2 x21)
0
1218  6 = 315
126 ÷ 21 = 6
-21
0
9
(2 x21)
(2 x21)
42
84
126
27
Level 4c
This must not be
introduced to the
children before
Year 5
18
18 ÷3 = 6
(1 x 3)
18
15
(1 x 3)
-
15
12
(1 x 3)
9
6
3 (1x3)
Understand division as
repeated subtraction
using a vertical line and
apparatus to make the
links
12
-
(1 x 3)
3 (1x3)
9
-
3 (1x3)
6
(1 x 3)
-
3
3 (1x3)
3
(1 x 3)
0
3 (1x3)
-
3 (1x3)
0
28
Level 4b
Year 5 and 6 only once confident
with previous method
What facts do I
know about the
7 times-table?
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
1x7=7
518 ÷ 7 = 74
10 x 7 = 70
- 350 ( 50 x 7 )
100
20 X 7 = 140
168
( 10 x 7 )
- 140
30
- 28
5 x 7 = 35
518
100 ÷ 7 = 14 r 2
- 70
2 x 7 = 14
50 x 7 = 350
( 20 x 7 )
100 x 7 = 700
28
(4x7)
- 28
2
(4x7)
0
Level 4a
Year 6 Top set only
560 ÷ 24
23 r8
24
56 0
Standard written method
Links directly to large
chunk subtraction
48 0
8 0
7 2
8
29