Transcript here

÷
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 and images to reinforce understanding.
When teaching a new strategy it is important to start
with numbers that the child can easily manipulate so
that they can understand the concept.
The transition between stages should not be hurried
as not all children will be ready to move on to the next
stage at the same time, therefore the progression in
this document is outlined in stages. Previous stages
may need to 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
written methods
for Addition
Number Track
Number Line
Expanded method
Partitioning and recombining
Formal Compact Method
3
4
Stage 1 – Number Track
add 3
0
1
2
3
and
+
4
5
6
7
8
9 10
7 + 3
and
• understand addition is combining
groups of objects
• count on using a number track
• use a puppet to accentuate jumps
+ =
+
=
•understand addition can be done in5
any order
Stage 2a – Introducing the number line
+ 3
0
1
2
3
4
0
1
2
3
4
and
+
5
5
6
6
7
7
8
8
9
9
10
10
7 + 3
and
• link number track and number line
• count on using a fully numbered
number line (start counting on in ones
6
and then move on to larger jumps)
Stage 2b – Using a number line
Partition the
smallest number.
Add the unit(s) first
13 + 11
13
+ 1
+ 10
14
24
13 + 18
+ 7
13
+ 1
20
+ 10
21
31
• always encourage ESTIMATION first
• teach and encourage children to
partition numbers in different ways in
order to bridge to the nearest multiple of 10
• start from the largest number and then count on
• progress from not bridging 10 to bridging through 10
7
• progress from fully numbered line to partially
numbered line then blank
Stage 3 – Partitioning & Recombining
The Expanded Method
40
20
3
4 0
+
3
2 0
+
8
6 0
+
1 1
• Encourage ESTIMATION
=
8
7 1
• reinforce place value by using place value cards to
partition alongside place value apparatus (Dienes)
• reinforce ‘carrying’ use equipment alongside
expanded written method to bridge from concrete
to abstract (see appendix a for recording –
8
transition
between equipment – pictorial recording and then
Stage 4 – Expanded Method leading
to Formal Compact Method
40
20
3
8
+
4 0
+
3
2 0
+
8
6 0
+
1 1
=
7 1
Add the unit(s) column then the ten(s)
column to calculate the final answer
4 3
+
2 8
7 1
1
Expanded
method leading
to compact
method
9
Stage 4 – Expanded Method leading
to Formal Compact Method (decimal)
0.40
0.20
0.03
0.08
+
0.40
+
0.03
0.20
+
0.08
0.60
+
0.11
=
0.71
• continue to encourage ESTIMATION
• link to money (add more than two amounts) & measurement
• link to using a calculator/interpreting calculator display
0.43
+ 0.28
0.71
1
C
M
√
AC
C
%
±
÷
7
8
9
x
4
5
6
-
1
2
3
0
.
=
+
Remember to line up
decimal points especially
when number of
10
digits differs
Progression in
written methods
for Subtraction
Number Track
Number Line
(Finding the difference
and counting back)
Expanded method
Partitioning and recombining
Formal Compact Method
11
12
Stage 1 – Number Track
take away 3
0
1
2
3
4
5
7
3
seven
three
6
7
8
9 10
• understand subtraction is taking
away objects
• jump/count back along a number track
• use a puppet to accentuate jumps
13
Stage 2a – Introducing Number Line
0
1
2
3
4
5
6
7
8
9 10
7
8
9
- 3
0
1
2
3
4
7
3
seven
three
5
6
10
• link number track and number line
• understand subtraction is taking away
objects
• jump/count back along a fully numbered
number line (start counting back in ones
14
and then move on to larger jumps)
Stage 2bi – Using a number line
(not bridging through 10)
23 – 18
finding the
difference
5
18
counting back
23
37 – 13
• Partition the smallest number.
Count back the units, then tens.
- 3
- 10
24
34
37
• ESTIMATE first
• understand ‘finding the difference’
AND ‘counting back’ has the same result
• promote finding the difference when the
numbers involved are close together
• progress from counting back in ones to 15
larger steps.
Stage 2bii – Using a number line
(bridging through 10)
43 – 27
- 10
16
- 4 - 3
- 10
26
36
40
43
• ESTIMATE first
• encourage children to partition numbers in
different ways
• bridge through multiples of 10
• ensure children have the opportunity to
solve subtraction problems in a range of
different contexts
16
• encourage use of vocabulary and explanation
Stage 3a – Expanded Method
(no exchanging)
47 - 14 = 33
take away the units
and then take away
the tens
40
7
- 10
4
30
and
3
• use place value apparatus (Dienes) to
re-inforce concept of exchanging
• move from concrete apparatus to
expanded written method (see appendix a for
recording – transition between equipment – pictorial
17
recording and then abstract)
• continue to encourage ESTIMATION
Stage 3b – Expanded Method
(with exchanging)
43 - 27 = 16
to subtract 7
units we need to
exchange a ten
for ten units
30
10 +
40
3
- 20
10
7
and
6
• use place value apparatus (Diennes) to
re-inforce concept of exchanging
• move from concrete apparatus to
expanded written method (see appendix b for
recording – transition between equipment – pictorial
18
recording and then abstract)
• continue to encourage ESTIMATION
Stage 4a – Formal Compact Method
30
10 +
40
- 20
3
7
10
and
3
6
1
4 3
- 2 7
1 6
• move from expanded written method
to compact method
19
• continue to encourage ESTIMATION
Stage 4 – Formal Compact Method
(decimal)
1
4.3
3
Remember to line up
decimal points especially
when number of
digits differs
- 2.7
1.6
• continue to encourage ESTIMATION
• link to money (giving change) and
measurement
• link to using a calculator/interpreting
calculator display
C
M
√
AC
C
%
±
÷
7
8
9
x
4
5
6
-
1
2
3
0
.
=
+
20
Progression in
written methods
for multiplication
Repeated addition, arrays
Grid method
(with imagery)
Grid method
Long multiplication
21
22
Stage 1 – Repeated addition, arrays
2+2+2+2=8
4x2=8
2 multiplied by 4
4 lots of 2
• understand that multiplication is a
shortened form of repeated addition
• understand multiplication as arrays23
and jumps on a number line
Stage 2 – Modelling grid method with
place value equipment
4 x 13
‘four lots of thirteen’
10
3
4
4
10
3
40
12
40 + 12 = 52
• use place value apparatus to illustrate
grid method, encourage jottings
• use digits of 5 and below to avoid
‘difficult’ tables and ensure method is secure
24
Stage 2a – Modelling grid method with place
value equipment (multiples of 10)
4 x 23
‘four lots of twenty three’
20
3
4
20
4
( 2 x 10 )
3
80
12
(4 x 2 x 10)
(4 x 3)
80 + 12 = 92
• use place value equipment to illustrate
grid method with multiples of 10
• reinforce using known facts to
25
multiply e.g. 4 x 20 = 4 x 2 x 10
Stage 3 – Grid method (no apparatus)
45 x 6
40
6
7
6
240
36
(6 x 4 x 10)
(6 x 6)
240 + 36 = 276
47 x 52
40
( 4 x 10 )
50
2000
2
80
(4 x 5 x 10 x 10)
(4 x 2 x 10)
80
350
(7 x 5 x 10)
14
(7 x 2 )14
2000
350
80
+ 14
2444
•continue to reinforce using known
facts to multiply e.g. 40 x 50 = 4 x 5 x 10 x 10
•progress to using the grid method
26
efficiently to multiply decimals
Stage 4 – Long multiplication
5 6
× 2 7
1 1 2 0
3 9 2
1 5 1 2
4 1
(56 × 20)
(56 × 7)
1
• ONLY move on to this method if
understanding of grid method is
secure
27
Stage 4a – Long multiplication
(decimal)
5.6
× 2.7
11.20
3.92
15.12
4 1
(5.6 × 2.0)
(5.6 × 0.07)
1
• continue to encourage ESTIMATION
(re-inforce place value)
• link to money and measurement
• link to using a calculator and
interpreting display
C
M
√
AC
C
%
±
÷
7
8
9
x
4
5
6
-
1
2
3
0
.
=
+
28
÷
Progression in
written methods
for division
÷
Division as sharing and grouping
Grouping on a number line
Link division and multiplication
Vertical recording
Chunking (fact box)
Short/long division
29
30
Stage 1 – Division as sharing and
grouping
sharing
one at
a time
÷
15 divided into
3 equal groups
÷
15 divided into
5 equal groups
understand division as sharing, understand
division as grouping
31
• understand remainders
•
Stage 2 – Grouping on a number line

• understand that division is
repeated subtraction
• show division as equal groups on a
number line
32
• then begin to understand remainder
Vertical recording (teacher model only)
0
3
6
9
12
18
18 ÷3 = 6
18
- 3
15
- 3
12
18
-
9
-
3
( 1 x 3 )
6
- 3
- 3
3 ( 1 x 3 )
9
- 3
3
3 ( 1 x 3 )
1 2
-
6
3 ( 1 x 3 )
1 5
- 3
0
15
3
( 1 x 3 )
3
-
3 ( 1 x 3 )
0
•turn horizontal number line vertical so children can
33
see link to vertical calculation and model recording,
use to illustrate need to take ‘chunks’ for efficiency
Stage 3 – Linking division & multiplication
leading to chunking
– introducing fact box
96  5
96 ÷ 5 = 19 r 1
96
- 50
( 10 x 5 )
Fact Box
46
- 25
What facts
do I know
about the
5 timestable?
( 5 x 5 )
1 x 5 = 5
21
2 x 5 = 10
- 20
5 x 5 = 25
1
10 x 5 = 50
• children need to see that when numbers are larger
it is more efficient to subtract larger ‘chunks’
• building a fact box will help children with the size
of the ‘chunks’
• children need to work with and without remainders
considering if answer needs rounding up or rounding
34
down
Stage 4 – Chunking with a fact box
100 ÷ 7 = 14 r 2
100
- 70
( 10 x 7 )
30
- 28
( 4 x 7 )
What
facts
do I know
about the
7 timestable?
2
518 ÷ 7 = 74
1 x 7 = 7
518
- 350
( 50 x 7 )
168
- 140 ( 20 x 7 )
28
- 28
0
Fact Box
( 4 x 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
• children need to see that when numbers are larger it is
more efficient to subtract larger ‘chunks’
• building a fact box will help children with the size of the
‘chunks’
35
• children need to work with and without remainders
considering if answer needs rounding up or rounding down
Stage 5 – Long division
560 ÷ 24
2 3 r 8
2 4
5 6 0
- 5 5 2
8
Jottings – Fact Box
x
20
4
20
400
80
3
60
12
400 + 80 + 60 + 12 = 552
• ONLY move on to this method if
understanding is secure
• move on to show remainders as a
fraction and decimal
36
Stage 5a – Long division
(showing remainder as a fraction)
560 ÷ 24
2 3 r 8/24 ()
2 4
5 6 0
- 5 5 2
8
Jottings – Fact Box
x
20
4
20
400
80
3
60
12
400 + 80 + 60 + 12 = 552
• ONLY move on to this method if
understanding is secure
• move on to show remainders as a
fraction and decimal
37
Stage 5b – Long division
(showing remainder as a decimal)
560 ÷ 24
2 3.333
2 4
5 6 0.00
- 5 5 2
8 0
7 2
8
Jottings – Fact Box
x
20
4
20
400
80
3
60
12
400 + 80 + 60 + 12 = 552
• ONLY move on to this method if
understanding is secure
• move on to show remainders as a
fraction and decimal
38
12 + 19
T - tens
U - units
Start with
apparatus then
show children how
to record pictorially
12 + 19
39
19 - 12
T - tens
U - units
Start with
apparatus
then show
children how
to record
pictorially
19 - 12
40
Link division and multiplication
12 divided
into
groups of 3 gives 4
groups
12  3 = 4
12 divided
into groups of 4
gives 3 groups
12  4 = 3
3 x 4 = 12 or 4 x 3 = 12
12  4 = 3 or 12  3 = 4
• understand that division is the
• inverse of multiplication
• reinforce division as grouping
• emphasise link between times table facts 41
and division facts
Understanding the inverse/finding
unknowns (empty boxes)
I can work out missing numbers in a number
sentence (year 1 & Year 2)
•
Introducing the Inverse – Play Mrs/Mr Opposite. Every
instruction the teacher gives the children have to do the
opposite e.g. teacher says take one step forward,
children take one step backwards or teacher says turn to
the right, children turn to the left etc. Explain that in
maths we call the opposite the inverse and that we are
going to be looking at the inverse (opposite) of adding.
•
Addition and Subtraction – Numicon Families (Using the
inverse) Ten is the same as/equals nine add/plus one.
Explain that the children are going to be using Numicon.
Show them what it is and explain how it is structured.
Model how this can be used to demonstrate the inverse
•
Ten is the same
as/equals nine
add/plus one.
10 = 9 + 1
•
Once imagery is secure replace one piece of Numicon
with an empty box. Remember to move the = sign!
42
Understanding the inverse/finding
unknowns (empty boxes)
I can work out missing numbers in a number sentence
including where = sign is used to
balance an equation
e.g. 6 + 4 = 3 + ? or 6 x 4 = 3 x ? (Year 3)
•
Use Numicon and balance to model and for the children
to practise in order to reinforce understanding of
equality.
•
Once imagery is secure replace one piece of Numicon
with an empty box. Remember to move the = sign AND
begin to explore balancing different operations.
•
I can work out missing numbers in a number sentence
including those where = balances an equation e.g.
10 – 3 = 3 + ? or 2 x 3 = 60  ? (Year 4)
Again use Numicon. Emphasis on exploring balancing
equations with different operations.
I can work out missing numbers in more complex
calculations e.g. 3?67 – 192? = 1539 or
32500  ? = 325 (Year 5)
I can find the unknown in a calculation such as
0.215 + ? = 0.275 or 5.6  ? = 0.7, drawing on
knowledge of number facts and place value,
including using a calculator and inverse operations
to solve more complex problems such as 568.1 43
 ?
= 24.7 and explain my reasoning (Year 6)