Threshfield Primary School
WHOLE SCHOOL PROGRESSION IN
CALCULATION STRATEGIES IN
MATHEMATICS
GUIDANCE FOR
PARENTS & FAMILIES
Monday 22nd September 2014
1
Stage
1
2
3
4
5
6
Addition
Subtraction
Multiplication
Division
Use of practical apparatus to
find the total.
Recording and developing
mental pictures.
Use of practical apparatus to
take away or find how many
more.
Recording and developing
mental pictures.
Progression in the use of a
number line.
The empty number line as a
representation of a mental
strategy.
Practical equipment using
exchange to take away.
Use of practical apparatus to
find the total of groups of
the same size.
Recording and developing
mental pictures.
The bead string, number
line and hundred square.
Arrays.
Use of practical apparatus to
share and put into equalsized groups.
Recording and developing
mental pictures.
The bead string, number
line and simple multiples.
Arrays for division.
The grid method.
“Short” division
Making the link between
the practical and columnar
subtraction.
Compact method.
Expanded “short”
multiplication.
“Long” division
Progression in the use of a
number line.
The empty number line as a
representation of a mental
strategy.
Partitioning into 10s and 1s
to lead to a formal written
method.
Using Diennes / place value
counters alongside
columnar written method.
Compact column method.
a)
b)
c)
Short multiplication
for TU and HTU
multiplied by any
number up to 12.
Expanded long
multiplication.
Compact long
multiplication.
2
ADDITION
3
Stage 1:
Use of practical apparatus to find the total.
Recording and developing mental pictures.
and
2
+
3
=
5
4
Stage 2:
Progression in the use of a number line
8 + 5 = 13
5
Stage 3:
The empty number line as a representation of a mental
strategy.
34 + 23 = 57
6
Stage 4:
Partitioning into tens and ones to lead to a formal written
method.
25 + 47
= 72
7
Written partitioning:47 + 25
40 + 20 = 60
60 + 12 = 72
and 7 + 5 = 12
Or
47 + 20 = 67
67 + 5 = 72
8
Stage 5
Using Dienes apparatus / place value counters alongside
columnar written method.
25 + 47 =
9
Stage 6:
Compact column method
10
SUBTRACTION
11
Stage 1:
Use of practical apparatus to take away or find how many more.
Recording and developing mental pictures.
12
Stage 2:
Progression in the use of a number line.
8–3=5
13
Additional ‘number lines’ - the bead string and hundred square
14
Stage 3:
The empty number line as a representation of a mental strategy.
Finding an answer by COUNTING BACK –
representing taking away.
15 – 7 = 8
15
74 – 27 = 47
“Take away 20”
“Take away 7”
16
Finding an answer by COUNTING ON –
representing the “difference between”.
17
Stage 4:
Practical equipment using exchange to ‘take away’.
72 – 47
72 – 47 = 25
18
Stage 5:
Making the link between the practical and columnar subtraction.
72 - 47
19
Stage 6:
Compact method
20
MULTIPLICATION
21
Stage 1:
Use of practical apparatus to find the total of groups of the
same size.
Recording and developing mental images.
2 + 2 + 2 + 2 + 2 = 10
5 + 5 + 5 + 5 + 5 + 5 = 30
6 x 5 = 30
22
3 lots of 4 are 12
4 lots of 3 are 12
23
Stage 2:
The bead string, number line and hundred square.
24
Stage 3:
Arrays
25
26
27
Stage 4:
The Grid Method
7 x 8 = 25 + 15 + 10 + 6 = 56
28
4 x13
= 52
This then becomes:
x
4
10
40
3
12
40 + 12 = 52
29
18 x 13
100 + 80 = 180
30 + 24 = 54
180 + 54 = 234
30
53 x 16
Estimate 50 x 20 = 1000
x
50
3
10
500
30
6
300
18
Adding the rows is the most efficient calculation:
500 + 300 = 800
30 + 18 = 48
So
800 + 48 = 848
31
Stage 5:
Expanded short multiplication
38 x 7
X
30 + 8
7
56
210
266
(7 x 8)
(7 x 30)
32
Stage 6:
Short multiplication for up to TU x 12
33
53 x 16
53
x 16
18
300
30
500
848
(6 x 3)
(6 x 50)
(10 x 3)
(10 x 50)
34
35
DIVISION
36
Stage 1:
Use of practical apparatus to SHARE and put into EQUAL-SIZED
GROUPS.
Recording and developing mental images
37
Stage 2:
Bead strings, number lines and simple multiples.
15 eggs are placed in baskets, with 3 in
each basket.
How many baskets are needed?
38
Counting on a labelled and then blank
number lines.
15 ÷ 3 = 5
39
Stage 3:
Arrays for division
56 ÷ 7
Dividend (56) ÷ divisor (7) = Quotient (8)
40
Stage 4:
Short division
Just as multiplication is repeated addition, so
division is repeated subtraction,
i.e. taking away the same number repeatedly.
432 ÷ 5
432
200
232
200
32
30
2
(40 x 5)
(40 x 5)
(6 x 5)
“CHUNKING” –
initially, this could
be done on an
open number
line.
How many 5s have been subtracted in total?
Answer = 86, with 2 remaining.
41
42
Stage 5
Long division
43
44
Notes:
We teach the children to ask themselves 4 questions – in steps about the calculations they are doing:1. Is it a calculation you can do in your head (mentally)? If yes,
then do it mentally. If not, then …
2. Is it a calculation you can do with jottings? If yes, then do it
using jottings. If no, then …
3. Is it calculation where you need a more formal written
method? If yes, then choose the appropriate method. If no,
then …
4. Is it a calculation where you need a calculator? If yes, then use
a calculator.
45
There are 7 strands of mathematical learning:






Using & applying maths
Counting & understanding number
Knowing & using number facts
Calculating
Understanding shape
Measuring
Handling data
Calculating is just one of those strands
46
Using &
Applying Maths
Calculating
Knowing number
facts
Understanding
place value
47