Transcript Progression in Calculations Written methods of

÷
Abbey Primary School – December 2012
1
Introduction
Written methods of calculations are reliable and
efficient procedures which can be used in many different
contexts. However, they are of no use to someone who
applies them inaccurately or who cannot judge whether
the answer is reasonable.
The progression towards a written method is crucial,
since it is based on steps which are done mentally and
which need to be secured first.
A structured approach to developing written methods
relies on:
1) Apparatus
2) Apparatus + picture
3) Apparatus + picture + recording
4) Picture + recording
5) Recording
Developing a common whole school approach enhances
pupils’ confidence and aids continuity across the school.
It can be hard for a pupil to hold all the steps of a
calculation in their heads, so informal jottings should be
encouraged at all times.
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.
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 and Images
Counting apparatus
Place value apparatus
Place value cards
Number tracks
numicon
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
40
8
Key Vocabulary
add
addition
plus
and
count on
greater
more
Most
sum
total
altogether
increase
3
Recognise numbers 0 to 10
1, 2, 3, 4, 5, 6
… there are 6
teddies
Find one more than a number
Cardinality: Knowing that 6 objects is
the number 6.
Count reliably up to 10 everyday objects
One more than
three is four
Count in ones and tens
Count all
3+2=5
Begin to use the + and = signs to record
mental calculations in a number sentence
Count on (keeping number in
your head / hiding the first
pattern)
6 + 4 = 10
4
Know doubles of numbers and their
corresponding halves
Know by heart all pairs of numbers
with a total of 20
3 7
Know that addition can be
done in any order
Put the biggest
number first and
count on
Begin to partition and
recombine numbers in
order to add
3+5
+3
5
8
5
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
25
35
15
16
17
18
25
26
27
28
Adding two two-digit numbers
(without bridging)
Counting in tens and ones
15 + 13 = 28
15
25
28
15 + 10 + 3 = 28
+30
Adding two two-digit numbers
(bridging through tens boundary)
Using a number line
48 + 36 = 84
48
+2
78
+4
80
84
48 + 30 + 6 = 84
6
40
Adding two two-digit numbers
(bridging through tens boundary)
8
30
6
Using place value cards and place
value apparatus to partition numbers
and recombine
48
48 + 36 = 84
+ 36
14 (8 + 6)
Remember to add the least
significant digits (e.g. units
70 (40 + 30)
in this case) first
84
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 = 84
T
U
40 + 8
+
30 + 6
80 + 4
10
7
Standard written method
The previous stages reinforce what
happens to the numbers when they
are added together using more
formal written methods.
48
+ 36
84
1
225 + 156 = 381
200 + 20 + 5
+ 100 + 50 + 6
300 + 80 + 1
225
+1 5 6
3 81
1
10
8
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 and Images
Counting apparatus
Place value apparatus
Place value cards
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
40
8
Key Vocabulary
Subtract
take away (specific use)
minus
count back
less
fewer
difference between
9
Begin to count backwards in
familiar contexts such as
number rhymes or stories
Five fat sausages
frying in a pan …
Ten green bottles
hanging on the wall
…
Continue to count back in
ones from any given number
Take away
Three teddies take
away two teddies
leaves one teddy
Find one less than
a number
Count back in tens
If I take away four shells
there are six left
Count backwards
along a number line10
to ‘ take away
Maria had six sweets and
she ate four. How many
did she have left?
Begin to use the – and = signs
to record mental calculations in
a number sentence
6-4=2
Know by heart subtraction facts
for numbers up to 10 and 20
Begin to find
the difference
40 – 27 = 13
+1
+1
27 28
+1
29 30
Find the difference using
the number line
+ 10
40
11
Find the difference using
the number line
40 – 27 = 13
+ 10
+3
27
30
Find the difference using
the number line
40
74 - 27 = 47
Extend to 3 digit numbers and decimals
12
column subtraction (without exchanging)
43 – 32 = 11
43
-32
11
40 + 3
- 30 + 2
10 + 1
Extend to 3 digit numbers without exchanging.
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
column subtraction (with exchanging)
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
10 +
3
4 13
- 2 7
1 6
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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
Models and Images
Counting apparatus
Place value apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Counting stick
Bead strings
Models and Images charts
ITPs – Multiplication grid, Number Dials, Multiplication Facts
40
8
Vocabulary
lots of
groups of
times
multiply
multiplication
multiple
product
once, twice, three times
array, row, column
double
repeated addition
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Count in tens
from zero
0
10
20
30
40
50
Count in twos
from zero
0
2
4
6
8
10
Count in fives
from zero
0
5
10
15
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
15
Use known facts to
work out new ones
Understand multiplication
as repeated addition
2+2+2+2=8
4x2=8
2 multiplied by 4
4 lots of 2
Understand
multiplication
as an array
Understand how to
represent arrays
on a number line
16
10
Use place value apparatus to support
the multiplication of U x TU
4
13 x 4 = 52
10
4
40
4
3
12
10
3
40
12
Use place value apparatus to support
the multiplication of U x TU
alongside the grid method
13 x 4 = 52
40 + 12 = 52
10
Use place value apparatus to
represent the multiplication
of U x TU alongside the grid
method
23 x 4 = 92
3
10
3
4
4
20
3
80
12
80 + 12 = 92
17
Multiplying TU x TU
33 x 14 = 462
30
3
10
300
30
120
4
120
12
30
300
+ 12
462
Use same method to multiply HTU X U
18
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
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Models and Images
Counting apparatus
Arrays
40
8
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
ITPs – Multiplication grid, Number Dials, Grouping, Remainders
Vocabulary
lots of
groups of
share
group
halve
half
divide
division
divided by
remainder
factor
quotient
divisible
19
Count on in tens
0
10
20
30
40
50
Count on in twos
0
2
4
6
8
10
5
10
15
20
25
Count on in fives
0
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
20
Understand division
as sharing
Understand division
as grouping
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
21
18 divided into groups of 3
Represent ‘groups’
for division on a
number line using
apparatus
alongside the line
18  3 = 6
0
1x3
3
6
1x3
0
3
9
1x3
6
12
1x3
9
15
1x3
12
18
1x3
18
15
Remember to ask the question – how many groups of 3 in 18?
18  4 = 4 r 2
Introduce remainders
1x4
0
1x4
4
8
r2
1x4
1x 4
12
16
18
How many groups of 4s in 18?
There are 4 groups of 4 with 2 left over.
22
Children need to see that as the
numbers get larger, chunking groups is
the more efficient method.
65  5 = 13
0
50
15
10 x 5
3x5
50
65
How many groups of 5 in 65?
There are 13 groups of 5 in 65?
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