÷
Introduction
The purpose of this policy is to outline and exemplify the strategies and methods
that we will teach children to enable them to become highly proficient and skilled
in mental and written calculations for the four operations of addition (+),
subtraction (-), multiplication (x) and division (÷). It will also show the progression
in skills and strategies through the maths curriculum. Written methods of
calculations are based on mental strategies and therefore children’s ability to
calculate mentally and use a range of known facts is given a high priority.
It is vital that children develop a strong and secure conceptual understanding of
the four operations. Therefore we use apparatus such as NUMICON, Base 10,
Cuisenaire and other relevant resources to support the development of this
understanding. We also use the connective model of maths so that children
understand the language, the images and models, the strategies and how to apply
them for each operation.
It is important that children are able to apply the most efficient and suitable
method for a given calculation and that they think like mathematicians, choosing
and applying the best strategies and estimating prior to calculating. They should
also check their calculations using the inverse.
Symbols
Language
Mathematical
image/picture
Context
Figure1.1 The connective model of learning mathematics
(adapted by the Devon Primary Maths Team)
The Importance of Mental Maths
Methods
•
•
•
•
It is vitally important the all children develop their ability
to calculate mentally using all four operations and are
able to draw upon their bank of known facts to support
them when calculating with larger numbers and decimal
numbers.
Learning addition and subtraction facts as well as been
able to count on and back in 1s, 10s, 100s, 1000s as
well as decimal steps from any start number are crucial
in supporting all mental calculations for addition and
subtraction. They also support the use of more formal
written methods.
Learning times table facts up to 12 x 12 and the
associated division facts provides children with the key
skills to enable them to multiply and divide both
mentally and using written methods. Understanding
what happens when a number is multiplied or divided
by 10, 100 and 1000, supports their understanding of
formal written methods.
Known facts need to be regularly practised so the
children can recall them at speed and with accuracy.
There is also a strong emphasis on teaching them to
utilise these facts to support other calculations.
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
• NUMICON
•Cuisenaire
•Base 10
• Place value apparatus
40
8
• Place value cards
• Number tracks
• Numbered number lines
• Marked but blank 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
Key Vocabulary
• add
• addition
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase
Recognise numbers 0 to 10
1, 2, 3, 4, 5, 6
… there are 6
teddies
Find one more than a number
Count reliably up to 10 everyday objects
One more than
three is four
Count in ones and tens
Begin to relate addition to
combining two groups of objects
3+2=5
Begin to use the + and = signs to record
mental calculations in a number sentence
Count along a number line to
add numbers together
6 + 4 = 10
Know doubles of numbers
Know by heart all pairs of numbers
with a total of 10 and 20
3 7
Know that addition can be
done in any order
3+5
Put the biggest
number first and
count on
4+6=1+9
4+6=1+9
Begin to partition numbers
in order to add
Understand ‘equals’ sign =
as ‘same as’
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
Adding two two-digit numbers
(without bridging)
Counting in tens and ones
Partitioning and recombining
15
+10
+3
25
15 + 13 = 28
28
Using place value cards and Base 10
to partition numbers and recombine
48 + 36 = 84
40
8
30
40 + 8
30 + 6
70 + 14 = 84
6
Using a number line
This is also a mental method which can be done mentally or written down. It
doesn’t matter how smaller number is added once it has been partitioned. I.e. It
can be added hundreds first or ones.
376 + 258 = 634
+ 50
+ 200
376
576
+8
626
634
Standard written method – Column Addition
The previous stages reinforce what happens to the numbers when
they are added together using more formal written methods. This
method can be used for any numbers and decimals.
48 + 36
48
+ 36
258
84
+ 376
1
634
1 1
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
NUMICON
Base 10
Cuisenaire
40
8
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
Key Vocabulary
Subtract
Take away
Minus
Count back
Less
Fewer
Difference between
Decrease
Five fat sausages
frying in a pan …
Begin to count backwards in
familiar contexts such as
number rhymes or stories
Ten green bottles
hanging on the wall
…
Continue the count back in
ones from any given number
Three teddies take away
two teddies leaves one
teddy
Begin to relate subtraction
to ‘ taking away ’
Find one less than
a number
Count back in tens
If I take away four shells
there are six left
4
5
6
7
8
9
10
Count backwards
along a number line
to ‘ take away
Count below the
number line when
counting back
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
45 – 1 =
Subtract 1 from a
two-digit number
-1
43
44
45
Subtract 10 from a twodigit number
-10
45 – 10 =
45
35
Subtract multiples of
10 from any number
45 – 30 =
-10
15
-10
25
-10
35
45
Begin to find the difference
by counting up from the
smallest number. This
method is absolutely
crucial in developing a
good understanding of
subtraction. It is also
the most efficient
mental method. It
should be used when
numbers are close
together.
Base 10 apparatus &
HTU grids can be
used as additional
support if necessary
74 - 27 = 47
Now where’s
the answer?
403 - 227 = 176
+3
227
+70
230
Standard written method
It is important that the children
have a good understanding of place
value and partitioning before using
this method. You don’t “borrow” or
“exchange”, you TAKE a ten or
hundred.
+3
+100
400
300
3
403
3
4 13
- 2 7
1 6
-
4 101 ¹3
2 7
1 8 6
Progression in Teaching Multiplication
Mental Skills
Recognise the size and position of numbers
Count on in different steps 2s, 5s, 10s
Double numbers 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
NUMICON
Cuisenaire
Place value apparatus
40
8
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
Vocabulary
Lots of
Groups of
Times
Multiply
Multiplication
Multiple
Product
Once, twice, three times
Array, row, column
Double
Repeated addition
Count in tens
from zero
0
10
20
30
40
Count in twos
from zero
0
2
4
6
8
Count in fives
from zero
0
5
10
15
20
Know doubles and
corresponding halves
Understand multiplication as
repeated addition
2+2+2+2=8
4x2=8
2 multiplied by 4
4 lots of 2
25
Know multiplication tables to 12X12 by
end of Year 4.
x5
2x5=10
3x5=15
6x5=30
8x5=40
Use known
facts to work
out new ones
10x5=50
Understand that …
24
x 20 = 24 x 2 x 10
24
x 50 = 24 x 5 x 10
Use factors to multiply
Understand multiplication
as an array
Understand how to
represent arrays
on a number line
2 hops of 4
4 hops of 2
4
2
4
2
2
2
13
Use place value
apparatus to support
multiplication
4
4 x 13 =
Multiplying TU x TU partitioning
the numbers
Multiplying U x TU
4 x 13 =
30
X
4
10
40
3
12
3
10
300
30
4
120
12
= 52
Multiplying TU x TU
14 x 33
= 330 +
= 132
462
Most children will use
this method. With this
method it is vital that
children make an
estimate of the answer
and can multiply by 10
and 100.
Formal Written Methods for
Multiplication:
Short Multiplication and Long
Multiplication
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
÷
Models and Images
NUMICON
Counting apparatus
40
8
Arrays
100 squares
Number tracks
Numbered number lines
Marked but unnumbered lines
Empty number lines.
Multiplication squares
Models and Images charts
ITPs – Multiplication remainders grid, Number Dials, Grouping,
Vocabulary
Lots of
Groups of
Share
Group
Divide
Division
Divided by
Remainder
Factor
Quotient
Divisible
Count back in tens
0
10
20
30
Count back in twos
2
4
6
8
10
Count back in fives
0
5
10
15
Know halves
Half of 6 is 3
½ of 6 = 3
Use known multiplication facts to work
out corresponding division facts


X
If 2 x 10 = 20
then
20  10 = 2
20  2 = 10
Understand division
as sharing USING
APPARATUS.
Understand division as
grouping USING
.
APPARATUS
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
9
12
15
18  3 = 6
18  6 = 3
82=4
Understand division as
repeated addition using
a horizontal line and
apparatus to make the
links
4 jumps of 2
+2
0
+2
2
+2
4
+2
6
8
18
Children need to see that as the
numbers get larger, large chunk
addition is the more efficient method.
Multiples of the divisor (large chunks)
are added. Multiplication facts are
needed to see the size of the ‘chunk’.
What facts do I
know about the
7 times-table?
Fact Box
518  7= 74
1x7=7
Estimate answer first
2 x 7 = 14
5 x 7 = 35
40x7
20x7
10x7
10 x 7 = 70
4x7
20 X 7 = 140
50 x 7 = 350
0
280
420
490
518
100 x 7 = 700
519  7= 74 r 1
40x7
0
20x7
280
10x7
420
490
4x7
r1
518
519
Formal Written Methods for Division
Short Division and Long Division