Transcript Progression in Calculations Written methods of calculations are

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1
Introduction
This policy contains the progression in calculation procedures that are
taught throughout Busbridge Infant School.
It emphasises the importance in calculating mentally as well as
focusing on writing procedures. Mental calculation is complementary to
written recording as in every written method there is an element of
mental processing. Written recording helps the children clarify their
thinking.
The understanding of the number system starts with counting. It is a
tool for calculation strategies. The key aspects of number are ‘nominal’
(number as labels), ‘ordinal’ (order of numbers and 1 to 1
correspondence) and ‘cardinal’ (the final number is how many in a set).
Each of the four operations (addition, subtraction, multiplication and
division) builds on mental skills e.g. number bonds, doubles, counting in
different steps etc, which provide the foundation for jottings and
informal written methods of recording.
Strategies for calculation need to be supported by familiar models,
images and practical activities to reinforce understanding.
A sound understanding of the number system is essential for children
to carry out calculations efficiently and accurately.
The long term aim is for children to be able to select the most
efficient method that is appropriate for a given task.
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Numbers as Labels and for Counting
5 currant buns…
Say some number names in familiar
contexts; such as nursery rhymes
1, 2, 3… there
are 3 teddies
11, 12, 13, 14,
15… there are
16 teddies
Count reliably up to 10 everyday objects
Count reliably up to
20 everyday objects
Say number names in
order
Recognise the numerals 1 – 9 then
10-20
Order and write numbers up to 10
then up to 20
Uses developing mathematical
ideas and methods to solve
practical problems
There are 5 spots on this
side and 5 on the other.
They are the same!
Calculating
Responds to the vocabulary
involved in addition and
subtraction in rhymes and games
Recognises the differences in
quantity when comparing sets of
objects
Finds 1 more or one less from a
group of up to 5 objects
5 little men in a flying
saucer…
This set has more
If 1 get 1 more cube
then there will be
1,2,3,4…4 cubes
Relates addition by combining
two groups
Relates subtraction to taking
away
Three teddies take
away two teddies
leaves one teddy
Calculating
We need 3 milks. We
have 4 so I need to
take one away
In practical activities
begins to use the
vocabulary involved in
adding and subtracting
Finds one more or one less than
a number to 10
Uses developing mathematical
ideas and methods to solve
practical problems including
sharing and halving
There are 10 bricks.
5 for you and 5 for me
Use a range of strategies
for addition and
subtraction including some
mental recall of number
bonds
Look 5 and 5 make 10.
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 using place value
• Bridge through 10
• Doubles and halves
Models, Images and apparatus
• Place value apparatus - Dienes
• Arrow/ place value cards
• Number tracks
• Numbered number lines
• Marked but unnumbered number lines
• Empty number lines
• Hundred square
• Counting stick
• Bead string
• Models and Images charts
Key Vocabulary
• add
• addition
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase
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8
Read and write numbers from 0
to 20 then to at least 100 in
numerals and in words
1, 2, 3, 4, 5, 6
… there are 6
cubes
Find one more/one less than a
number
Count reliably up to 10 then 20 everyday
objects
One more than
three is four
Begin to relate addition to
combining two groups of objects
Count in ones and tens
Begin to use the + and = signs to record
mental calculations in a number sentence
3+2=5
6 + 4 = 10
Count along a number line to
add numbers together
7
Put the biggest
number first and
count on
5+3
Know doubles of numbers
Know by heart all pairs of numbers
with a total of 10
3 7
Begin to partition numbers
in order to add
8
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
15
25
35
15
16
17
18
25
26
27
28
25
Know by heart all pairs of numbers
with a total of 20
28
Adding two two-digit numbers
Counting in tens and ones
Partitioning and recombining
15 + 13 = 28
13 7
6 and how many
more make 20?
9
Add near multiples of 10 to a 2
digit number by adding 10 and
adjusting
25 + 21 = 46
25+20+1=46
Add two single-digit
numbers that bridge 10
+2
8 + 7 = 15
8
8+2=10
+5
10
15
10+5=15
Adding two two-digit numbers
(bridging through tens boundary)
Using an empty number line
+30
48
OR
Using place value cards and Dienes
to partition numbers and recombine
48 + 36 = 84
48+30=78
78
+2
48
+2
80
84
+34
84
50
40
+4
8
30
6
78+6=84
40 + 30 + 8 + 6
40 + 30 = 70
8 + 6 = 14
70 + 14 = 84
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Progression in Teaching Subtraction
Mental Skills
•
•
•
•
•
•
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Recognise the size and position of numbers
Count back in ones and tens
Know number bonds for all numbers to 20 and related subtraction facts
Use the inverse of addition
Subtract multiples of 10 from any number
Partition and recombine numbers (only partition the number to be subtracted)
Bridge through 10
Doubles and halves
Models, Images and apparatus
Place value apparatus - Dienes
Arrow/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
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8
Key Vocabulary
Subtract
Take away
Minus
Count back
Less
Fewer
Difference between
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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 the count back in
ones from any given number
Begin to relate subtraction
to ‘ taking away ’
Three cubes take away
two cubes leaves one
cube
Find one less than
a number
Count back in tens
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
1
2
3
The
difference
between 8
and 5 is 3
Compare 2 sets to find
numerical difference
Understand the operation of
subtraction and use related
vocab. Subtract numbers when
solving problems involving up to
10 objects in a range of
contexts
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
7–2=5
Know by heart subtraction facts
for numbers up to 10
13
Begin to find
the difference
by counting up
from the
smallest
number
20
Subtract 10 from a twodigit number
30
Subtract multiples of
10 from any number
25
Begin to partition numbers in
order to take away
43 – 23
-3
20
45
- 10
23
43 –
- 10
33
20
3
43 – 20 = 23
43
23 – 3 = 20
14
Subtract single digit
numbers bridging through
10
15 - 7 = 8
15-5=10
10=2=8
Partition the number
to be subtracted
bridging through 10
74 - 27 = 47
74-20=54
54-7=47
Decide whether to
count up from the
smaller number or
count back depending
on the difference
between the two
numbers
74 - 27 = 47
74 - 67 = 7
Count back
Count up
Subtract mentally a near multiple of
10 from a 2 digit number
56 – 21 = 35
56-20=36
36-1=35
Know by heart subtraction facts
for numbers up to 20
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Progression in Teaching Multiplication
Mental Skills
Recognise the size and position of numbers
Count on in different steps 2s, 5s and 10s
Double numbers to 10
Recognise multiplication as repeated addition
Represent multiplication as arrays
Quick recall of multiplication facts in the 2, 5 and 10 times tables
Use known facts to derive associated facts
Multiplying by multiples of 10
Models, Images and apparatus
Place value apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines
Multiplication squares
Counting stick
Models and Images charts
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
20
30
40
50
Count in twos
from zero
0
2
4
6
8
Count in fives
from zero
0
5
10
15
20
25
Know doubles and
corresponding halves to 20
17
2+2+2+2=8
Understand multiplication as
repeated addition
4 x 2 = 10
2 multiplied by 4
4 lots of 2
Understand multiplication
as an array
Understand how to
represent arrays
on a number line as
repeated addition
2 x 5 = 10
Tables
spider
4 x 5 = 20
6 x 5 = 30
x5
10 x 5 = 50
Know by heart facts for the 2,5 & 10
multiplication tables
3 x 5 = 15
8 x 5 = 40
5 x 5 = 25
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Use known
facts to work
out new ones
Use known facts and
place value to carry out
mentally simple x
calculations
Know that multiplication of two
numbers can be done in any order
7 x 2=14
2 x 7=14
10
Use place value apparatus to support
the multiplication of U x TU
4 x 13=52
3
4
4x10=40
4x3=12
40+12=52
Use place value apparatus to support
the multiplication of U x TU
alongside the grid method
4 x 13
4
4
10
3
40
12
10
3
40
12
40 + 12 = 52
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Progression in Teaching Division
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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
Understand division as grouping or sharing
Quick recall of division facts for the 2, 5 and 10 times tables
Use the inverse of multiplication
Use known facts to derive associated facts
Divide by multiples of 10
Models, Images and apparatus
Counting apparatus
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines
Multiplication squares
Models and Images charts
40
8
Vocabulary
Lots of
Groups of
Share
Shared between
Group
Divide
Divide into
Division
Divided by
Remainder
Factor
Quotient
Divisible
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20
Give every
bear a sweet
– how many
sweets do we
need?
Solve problems by sharing
objects in a practical or role
play context
Count back in tens
0
10
20
30
Give each
bear 2
sweets – how
many sweets
do we need?
Solving practical
problems by sharing into
equal groups
Count back in twos
?
2
Count back in fives
4
6
0
8
5
10
15
21
Know halves to
20 and other
simple fraction
of numbers,
objects and
quantities
Half of 6 is 3
½ of 6 = 3
Quarter of 8 is 2
¼ of 8 = 2
Understand division
as sharing
1
2
3
4
6
7
8
9
5
10 etc….
Understand division
as grouping
1,2,3,4,5
Represent ‘groups’
for division as
repeated
subtraction on a
number line using
apparatus
alongside the line
to include
remainders
0
-3
18 divided into groups of 3
18  3 = 6
18  3 = 6
0
3
-3
3
6,7,8,9,10 etc
6
-3
6
9
-3
9
12
-3
12
15
-3
15
18
18 22