÷
September 2013
Aims
.
•To recap on the written
methods of calculations
using the four operations
• For teachers and
teaching assistants, to be
confident with teaching
the level of written
methods relevant to the
children they teach
Models and Images
Counting apparatus
Place value apparatus
Place value - arrow cards
Numbered number lines
Empty number lines
Hundred square
Counting stick
Bead string
Models and Images charts
40
8
Key
Vocabulary
Progression in Teaching Addition
Addition Skills
Recognise the size and position of numbers
Count on in ones and tens
Know number bonds to 10 and 20
Recall all addition pairs to 9 + 9
Add multiples of 10 to any number
Partition and recombine numbers
Bridge through 10
Add mentally a series of one-digit numbers
Level 1c
Begin to record their work ,
using own mark markings
Level 1b
Record their work with
objects, pictures and
diagrams. Begin to use the +
- signs
Level 1a
Jump in 1’s on a number line
Record additions using + =
The empty number line
or:
The empty number line helps to record the steps on the
way to calculating the total.
Steps in addition can be recorded on a number line. The
steps often bridge through a multiple of 10.
15 + 13 = 28
15
15
16
17
18
25
26
27
28
25
28
Level 2c
Jump in ones and
tens on a number
line to solve
problems
8 + 7 = 15
Level 2b
Using symbols + =
to write addition
sentences
48 + 36 = 84
Level 2a–
Record mental
calculations as
number sentences
Partitioning
The next stage is to record mental methods using
partitioning. Add the tens and then the ones to form
partial sums and then add these partial sums.
+30
48 + 36 = 84
48
78
+2
Adding two two-digit numbers
(bridging through tens boundary)
Using a number line
OR
48
+2
+4
80
+34
84
50
40
84
8
30
6
Using place value cards and place
value apparatus to partition numbers
and recombine
48 + 36 =
40 + 30 = 70
8 + 6 = 14
70 + 14 = 84
Level 3c –
Use informal written methods –
e.g number line
Use of partitioning to solve
addition problems
Expanded written method
Expanded method
T
U
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
70 + 14 = 84
Lots of practical, hands
on activities needed to
consolidate
understanding
Expanded method
used for
demonstration and
practical
work
1
Level 3b
add and subtract
numbers up to 3
digits using written
methods
Stage 4: Column addition
In this method, recording is reduced further. Carry
digits are recorded below the line, using the words
'carry ten' or 'carry one hundred', not 'carry one'.
Later, extend to adding three two-digit numbers,
two three-digit numbers and numbers with
different numbers of digits
Column addition remains efficient when used
with larger whole numbers and decimals. Once
learned, the method is quick and reliable.
Level 3a
Do column addition of numbers up to 1000
Level 4c
Do column addition of numbers up to 10 000
Level 4b
Use efficient short written methods with confidence
Level 4a
Use written methods
including decimals (up tp 2 d.p.)
1
Level 5 –
To revise use of written methods effectively to add
and subtract decimals.
Key Vocabulary
subtract
take away
minus
count back
less
fewer
difference between
Progression in Teaching Subtraction
Subtraction Skills
Recognise the size and position of numbers
Count back in ones and tens
Know number facts for all numbers to 10 and 20
Subtract multiples of 10 from any number (use
of hundred square)
Partition and recombine numbers (only partition
the number to be subtracted)
Bridge through 10
Level 1c
Begin to record their
work , using own
mark markings
Level 1b
Record their work
with objects,
pictures and
diagrams. Begin to
use the -
=
signs
Level 1a
Jump back in 1’s on
a number line
Record additions
using -
=
The empty number line
Steps in subtraction can be recorded on a number line. The
steps often bridge through a multiple of 10.
15 - 7 = 8
74 - 27 = 47 worked by counting back:
Level 2c
Jump back in ones
and tens on
number line
Level 2b
Use symbols to
record subtraction
number sentences
Level 2a
Record mental
calculations as
number sentences
using +, -, =
Counting on, counting back
Level 3c –
Use informal written methods, e.g. using
partitioning, a number line to solve add and
subraction
The Expanded method
Expanded method
43 - 27 = 16
to subtract 7 units
we need to exchange
a ten for ten units
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
T
- 2
U
10 +
3
7
Lots of practise with place value resources
To support and consolidate understanding.
Level 3b
Add and subtract numbers up to 100 using written
methods
Short Written Method
The previous stages reinforce
what happens to numbers
when they are subtracted
using more formal written
methods. It is important they
have a good understanding of
place value and partitioning
324
-178
-254x
Common
error
Use language
of exchange
Level 3a
Do column subtraction of whole numbers to 1000
Level 4c
Do column subtraction of whole numbers to 10 000
Level 4b
Use efficient written methods of add and sub
Level 4a
Use written methods for decimals
Level 5 –
To revise use of written methods
effectively to add and subtract
decimals.
Vocabulary
lots of
groups of
times
multiply
multiplication
Multiple - factor
product
once, twice, three times
array, row, column
double
repeated addition
Learning tables is the key!!
In year 6 all children should know all their
multiplication and division facts to 10 x 10 (level 4b)
http://www.maths-games.org/times-tables-games.html
Progression in Teaching Multiplication
Mental Skills
Recognise the size and position of numbers
Count on in different steps 2s, 5s, 10s
Double numbers up 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
Level 1c
Count repeated groups of the
same number
Level 1b
Count on or back in 1’s,
Level 1a
Count on or back in 2’s, 10’s
Level 2c
Begin counting in 5’s
Level 2b
Identify all doubles and halves
to 20
Level 2a
Know by heart facts for 2x and
10x
Count in tens
from zero
0
20
30
40
50
Count in twos
from zero
0
4
6
8
10
10
15
Count in fives
from zero
0
20
25
30
Know doubles and
corresponding halves
Know multiplication tables to 10 x 10
2 x 5 = 10
x5
6 x 5 = 30
3 x 5 = 15
8 x 5 = 40
Use known facts to
work out new ones
Understand that …
24
x 20 = 24 x 2 x 10
24
x 50 = 24 x 5 x 10
Understand multiplication
as repeated addition
Use factors to multiply
2+2+2+2=8
4x2=8
2 multiplied by 4
4 lots of 2
Understand
multiplication
as an array
Level 3c
Know by heart 2,5,10 x table
Solve one step simple multiplication problems
To use informal and practical methods to solve 2
digit x single digit; 4 x13
Level 4c
Use refined methods for TU xU
38 x 7
Level 4b –
Use efficient methods of HTU xU
Use written methods of TU xTU
Grid
Method
14 x 33
329 x 7
33
× 14
330
132
462
(33× 10)
(33 × 4
Long
multiplication
Level 4a/ Level 5
Use written methods of multiplication of numbers
including decimals
Vocabulary
lots of
groups of
share
group
halve
half
divide
division
divided by
remainder
factor
quotient
divisible
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÷
÷
Level 1c
Count back in ones
Share objects into TWO equal groups
and count how many in a group
Level 1b
Share objects into groups of 2, 10
Level 1a
Count back in 2’s.10,s
Level 2c
Count back in 5’s
Level 2b
Share objects into equal groups of
2,5,10
Identify halves to 20
Count back in tens
0
10
20
30
Count back in twos
?
4
6
8
10
Count back in fives
0
5
10
Know halves
Half of 6 is 3
½ of 6 = 3
Use known multiplication facts to work
out corresponding division facts
If 2 x 10 = 20
then
20  10 = 2
20  2 = 10
15
Understand division
as sharing
Understand division
as grouping
Level 2a
Begin to see division as an
array
Begin to understand
division as repeated
subtraction
Use the symbols ÷ and =
to record
and interpret number
sentences (e.g. □ ÷ 2 = 6
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
18 divided into groups of 3
Represent ‘groups’
for division on a
number line using
apparatus
alongside the line
18  3 = 6
0
3
6
18  3 = 6
0
9
12
18  6 = 3
15
18
18
Level 3c – use a number line to record division
18
18 ÷3 = 6
-3
18
15
-3
12
-
-3
3 (1x3)
9
-
6
3 (1x3)
6
-3
-
-3
Level 3b
Divide using a
number line to
support repeated
subtraction
/grouping
2 digit numbers by
2,3,4,5 and 10, wtih
whole number
answers and
remainders
3 (1x3)
3
3
0
3 (1x3)
12
-3
9
3 (1x3)
15
-
Understand division as
repeated subtraction
using a vertical line and
apparatus to make the
links
3 (1x3)
0
Record related
multiplication and
division number
sentences
Bus stop method
What facts do I
know about the
7 times-table?
Fact Box
Level 3a
Begin to use
informal methods
of division
e.g TU / U
1x7=7
2 x 7 = 14
3 x 7 = 21
4 x 7 = 28
5 x 7 = 35
Level 4c
Use refined
methods for
TU / U
Level 4b
use efficient methods for HTU / U
6 x 7 = 42
8 x 7 = 56
9 x 7 = 63
10 x 7 = 70
Level 4a
Use efficient methods for all four rules including
decimals
Level 5
To divide HTU / TU (long division)
Fact Box
1 x 21 = 21
2 x 21 = 42
3 x 21 = 63
4 x 21 = 84
5 x 21 = 105
6 x 21 = 126
7 x 21 = 147
8 x 21 = 168
9 x 21 = 189
10 x 21 = 210