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Calculations Policy
Multiplication and Division
Harbinger Primary School
September 2016
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Our aim is for each child to have an efficient strategy for each operation so that
they are able to calculate independently. This will be helped by all staff having a
shared understanding and common approach when teaching the methods.
The requirements of the new Maths curriculum for the 4 operations have been
set out for each year group, and ideas for areas of application have been included.
Teaching of mental maths, place value and number bonds is also vital, and regular reference to the inverse operation will strengthen understanding.
This can be particularly useful for checking that results are feasible, therefore, always ask children to write the calculation and answer horizontally
in order that they consider whether it makes sense as they finish their calculation. This also makes sure children are aware that a written
calculation is not always necessary.
With each new stage, revert to numbers that the children are comfortable with, and use ‘old’ and ‘new’ strategies side by side. When children are
feeling completely confident, calculations can become more complex and the new strategy used alone.
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Year 1
Curriculum learning objectives
• solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.
MULTIPLICATION
Notes
Pictorial representation:
(repeated addition)
3X2=6
2 +
2 + 2
Arrays:
3X2=6
DIVISION
Language: groups of, lots of
Pictorial representation:
Language: groups of, lots of, share
Areas for application:
Measurements
Geometry—how many sides altogether?
i)
Areas for application:
Measurements
Geometry—8 squares, how many sides?
6÷2=3
Dots/pictures/any representations
which children count to get the answer
= 6
Notes
Writing of number sentence follows as
child understands
x, / and = symbols
Arrays – More formal organisation/
accurate pictorial representation of
operation.
Either array is OK as long as the child’s
explanation represents the sum ie.
3 x 2 is 3 rows of 2 columns or 3 x 2 is
3 lots of 2 in a column. This also matches the spoken number sentence ie 3 lots
of 2
Dots/pictures/any representations
which children count to get the
Answer.
This will follow on from practical expe(Draw circles shown by the second number in the rience of sharing. It is necessary from
sum then ‘share’ all of the dots shown by the first this early stage to discuss the two
number)
different ways of understanding division ie i) 6 divided by 2 can be 6
divided into two groups (a logical proii)
gression from practical sharing) OR
ii) 6 divided into groups of 2 (most
helpful for future calculation strategies, esp no line)
“Two groups with three in each” OR
“Three groups of two”
Both are valid and should be recognised, with weighting on the latter
being increased to prepare the children
for moving forward
6÷2=3
(Draw the number of dots shown by the first
number in the sum then circle ‘groups’ of the
second number)
7÷2=3r1
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Year 2
Curriculum learning objectives
• recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers
• calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (×), division (÷) and equals (=) signs
• show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot
• solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts.
MULTIPLICATION
Pictorial representation
Arrays
Completed number line:
Notes
Language: groups of, lots of, times, multiply, multiplied by, multiple of, repeated addition
Areas for application:
Measurements
Geometry—how many sides altogether?
Fractions
Number lines always have to have the
‘how many lots of’ written above and
the size of the jumps inside.
Number lines are used as an aid to
counting the units inside the jumps of
‘lots of’
Start from 0
At first do the same number of jumps
as the 1st number in the sum.
Do the jumps the size of the 2nd number in the calculation . The answer is the
number the child lands on
DIVISION
Notes
Pictorial representation
Language: groups of, lots of, share,
divide by, divided by, remainder
Completed number line:
Areas for application:
Measurements
Geometry—8 squares, how many sides?
Fractions
Number lines always have to have
the ‘how many lots of’ written above
and the size of the jumps inside.
Use the first number in the sum to
mark where you will stop and jump in
lots of the second number. Count the
number of ‘lots of’ noted above the
jumps to reach the answer
Children may begin to use jumps of
more than one ‘lot of’ if they feel confident and can use their times table
knowledge to help them. This will be
helpful for the empty number line later
on.
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MULTIPLICATION
Empty number line:
Notes
Number lines always to how many ‘lots
of’ written above, the running total
kept along the bottom and the size of
the jump inside.
Children begin to realise they can start
with either number. This can be useful
in being more efficient ie in doing less
jumps or using known multiplication
facts.
“If I actually know what 3 lots of 4 are,
do I need to do a jump for each group or
could I do one big jump of the whole
amount?”
Children become more confident in
choosing the size of their jumps and
begin to use partitioning when ready
Children working at a lower level or with
SEN may use this strategy with a
multiplication square to support them
Children may do jumps within jumps if
they need a calculation to be simplified
Some children may use compensation,
particularly for calculation close to a
multiple of 10. This does not need to be
explicitly taught, but will be interesting
to recognise and discuss.
DIVISION
Empty number line:
Notes
Number lines always to how many
‘lots of’ written above, the running
total kept along the bottom and the
size of the jump inside.
Write the first number in the calculation on the right, zero on the left
At first jump in one ‘lot of’ until you
reach the right hand side of the number line
The answer is reached by counting how
many ‘lots of’ the second number were
jumped
Children then begin to use known times
table facts to do larger jumps ie more
than one ’lot of’ at a time. This is improving efficiency
“If I know that 2 lots of 3 are 6, can I
do this bigger jump? It will mean I can
write less and finish the sum quicker!”
“So 2 lots of 3 are 6, I think I can fit
in another 2 lots which will take me to
12. I don't think I can do another jump
this big, so if I try 1 lot of 3, where
does that take me to? I’ve landed on 15
which is the first number in my sum, so
if I count how many lots of I have
jumped altogether, there are 2, plus
another 2 is 4 and then add 1 more
group of 3 so my answer is 5.”
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Year 3
Curriculum learning objectives
• recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
• write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers,
using mental and progressing to formal written methods
• solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects
are connected to m objects.
MULTIPLICATION
Empty number line:
Notes
Language: groups of, lots of, times, multiply, multiplied by, multiple of, repeated addition, product of
Areas for application:
Measurements
Geometry—how many sides altogether?
Fractions
Discuss methods of multiplying by
10/100 accurately ie 70 x 8 can be
thought of as 7 x 8 = 56, then make 10x
bigger = 560 or we can count in tens of
7s: 70, 140, 210, 280, 350, 420, 480,
560
This strategy provides consolidates
understanding of the number system
and place value and allows children to
stretch themselves in these areas
DIVISION
Empty number line:
Notes
Language: groups of, lots of, share,
divide by, divided by, remainder
Areas for application:
Measurements
Geometry—8 squares, how many sides?
Fractions
As larger 2 digit numbers are used, the
size of jumps should continue to grow,
with jumps of 10 lots of being very
helpful.
The empty number line can be used for
3 and 4 digit numbers and beyond, using 10x, 100x and 1000x. Calculations
of this type are excellent for consolidating understanding of the number
system and place value, and are good
evidence that this is embedded. Methods of calculating 10x, 100x etc can be
discussed as noted in ‘Multiplication’
“I’d like to be able to do a big jump to
start my calculation off - I know that 7
lots of 6 are 42, 420 is ten times bigger so 70 lots of 6 equal 420. With one
more lot of 6 I land on 426, so the
answer is 71 with a remainder of 1”
Sums including remainders should be
used throughout
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MULTIPLICATION
Grid Method:
Notes
This provides a more formal way of
organising partitioning and builds on
existing place value knowledge
Emphasise need to present the
calculation carefully and consistently
Partition any number with 2 digits or
more.
Organise in grid, number with most
digits positioned horizontally.
Starting with the units, or the
‘outermost square’, work through the
individual calculations
To begin with, write the individual
calculations in the appropriate square
Add the totals together using their
current/preferred addition strategy ref to Addition & Subtraction
Calculation Policy
Children may use a number line or other
jottings alongside this strategy to keep
track of parts of the calculation
DIVISION
Notes
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Year 4
Curriculum learning objectives
• recall multiplication and division facts for multiplication tables up to 12 × 12
• use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together three numbers
• recognise and use factor pairs and commutativity in mental calculations
• multiply two-digit and three-digit numbers by a one-digit number using formal written layout (short division towards the end of Year 4)
• solve problems involving multiplying and adding, including using the distributive law to multiply two digit numbers by one digit, integer scaling problems and harder correspondence
problems such as n objects are connected to m objects.
MULTIPLICATION
Grid Method:
Notes
Language: groups of, lots of, times, multiply, multiplied by, multiple of, repeated addition, product of
DIVISION
Empty number line:
Notes
Language: groups of, lots of, share,
divide by, divided by, remainder, factor, divisible by
Areas for application:
Measurements
Geometry—how many sides altogether?
Fractions
Areas for application:
Measurements
Geometry—8 squares, how many sides?
Fractions
Most errors occur in the top left-hand
box and therefore discussion of place
value and the aforementioned methods
of multiplying accurately by
10/100/1000 remain very important.
The empty number line should be
used until a child demonstrates understanding and ability with the full
range of calculation shown in this
stage, after which they will move
onto compact division.
2 digit x 2 digit is the most complex of
these calculations and therefore most
open to error
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MULTIPLICATION
Expanded columns:
Notes
Numbers begin to be arranged in
columns, multiplication sign is to the
right and individual calculations are
written on the right.
DIVISION
Short division: (Summer of Y4?)
Read as “4 x 3, 4 x 10”
“3 x 4, 3 x 10, 3 x 200”
Crossing boundaries and moving into
other columns is good preparation for
compact method:
“4 x 3 is 12. Where should I write this?
Should both digits be under the 4 or do
I need to move into a different column?”
“6 x 20 is 120 so I will need to think
carefully about where I write that. How
will I know?” etc
Values to be carried across the columns
when adding the totals are noted below
the calculation, as in the Addition and
Subtraction Calculation Policy
Introducing this method using
dienes supports understanding and is
good provision for kinaesthetic and
visual learners. It will also be very
helpful when ‘moving’, ’taking’ or
’carrying’.
Emphasise the importance of vertical
organisation and place value
Begin with the units
“Does it matter which number I start
with? We know in multiplication sums
the answer will be the same no matter
which way the number sentence is ordered. It will really help our strategy to
begin with the number we are multiplying by, and make sure we are multiplying
it by every digit in the other number.”
Notes
“I have 3 Tens that need to be shared equally
into 3 groups. How many Tens does each group
get? They get 1 each—I’ll write that in the space
above, in the Tens column.
I have 6 Units that need to be shared equally into
3 groups. How many Units does each group have?
They get 2 each—I’ll write that in the Units column above. So 36 shared into 3 groups is 12.”
Individual digits should continue to be
spoken of in line with their true value,
with the emphasis being placed on
partitioning the first number in the
calculation in order to make the calculation manageable.
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MULTIPLICATION
Expanded columns:
Notes
2 digit x 2 digit is the most
complex of these calculations and therefore most open to error. Reverting to
the grid method to check the answer is
an option, and it provides an opportunity
for children to ensure they have included each of the 4 multiplications involved
DIVISION
Notes
“I need to share 72 into 6 groups. I
can put a rod of Ten in each group and
I can show this by writing 1 in the Tens
column. I have one Ten left over which
I can’t share equally as it is, so I will
need to break it into individual dienes
or ’ones’ or Units. One Ten is the same
as 10 Units. So now I have 12 Units
altogether which I’m going to make a
note of in the Units column. I can then
put 2 Units into each group and I will
write this answer in the Units column.
So 72 shared into 6 groups is 12.”
“I know that that the 8 means 8 Hundreds as I’m confident with place value,
so I’ll just think about it as an ‘8’.
How many groups of 4’s in 8? There are
2.
How many 4’s in 9? There are 2, but
one is left. I will move this to the Units
column—so there are now 12 Units
(because 1 ten makes 10 units).
How many 4’s in 12? There are 3.
So my answer is 223.”
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Year 5
Curriculum learning objectives
• identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
• know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
• establish whether a number up to 100 is prime and recall prime numbers up to 19
• multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
• multiply and divide numbers mentally drawing upon known facts
• divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
• multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
MULTIPLICATION
Short multiplication:
Notes
Language: groups of, lots of, times, multiply, multiplied by, multiple of, repeated addition, product of, square
Areas for application:
Measurements
Geometry
Fractions, decimals, percentages
Values being carried across columns
are noted below the calculation, with
the x sign on the right
Focus on becoming more efficient in
recording eg “This feels like I’m doing a
lot of writing, I wonder if I can cut out
some steps now that I am more confident and use what I know about carrying
into other columns from my addition
strategies?”
For an example explanation of the
strategy to the children, see over.
DIVISION
Short division:
(see Year 4 for the introduction of
short division)
Notes
Language: groups of, lots of, share,
divide by, divided by, remainder, factor, divisible by
Areas for application:
Measurements
Geometry
Fractions, decimals, percentages
“I know that the 4 means 4 Hundreds
as I’m confident with place value, so I’ll
just think about it as a ‘4’. How many
groups of 4’s in 4? There is only 1.
How many 4’s in 8? There are 2.
How many 4’s in 3? There aren’t any, so
I write 0 above. But I MUST note down
how many remain. As there aren’t any
groups of 4 in 3, it means that 3 remain / are left. I write this as r.3.
So my answer is 120 with 3 remaining.”
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MULTIPLICATION
Short multiplication:
Notes
Begin with examples such as 13 x 6 that
give an answer below 100. Progressing
onto larger values such as 28 x 6 then
requires a move into the hundreds column.
“6 x 3 is 18 so if I put the 8 where it
belongs in the units column, where could
I place the 10 to make sure it is included? I could move it underneath the total
line just like when I am doing compact
addition (if children are using compact
multiplication it can be expected that
they will be using compact addition)
Then is 6 x 10 is 60, how should I write
this? Do I need the 0? Have I got just 1
tens or are there any other tens I need
to include in this column?” etc.
Revert to previous strategies to consolidate/ ’prove’ if necessary.
This is also helpful for checking an answer as, with calculations of this complexity, using the inverse to check is not
feasible eg 1242 ÷ 27!
Discussion of 0 as a placeholder is very
important here as children may feel it
can be ignored eg “3 x 0 is 0, so can I
just move straight on? Do I need to
write 0? What happens to my answer if
I ignore it? 3 x 300 is 900. If I put this
next to the 6 it is worth 90 which is not
correct. 96 is not a reasonable answer
when multiplying 300 and something by
3. If I put it in the hundreds column
then there is a gap in my answer, so that
0 is important to hold in the tens column
to make sure my answer is 906.”
Accurate vertical alignment of decimal
point as part of place value is crucial
here. 1 decimal place is sufficient but 2
decimal places may be used if working
within the context of money.
Long multiplication—children can check
answers using the grid method.
DIVISION
Short division:
Notes
“How many 7’s in 6? There are none, so
we use both the 6 AND the next digit
0. Therefore my calculation becomes:
how many 7’s in 60? There are 8 with 4
remaining.
How many 7’s in 42? There are 6.
Therefore my answer is 86.
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MULTIPLICATION
Long multiplication:
Notes
For calculations x 2-digits
It is VERY important to
discuss 0 as a placeholder.
“We know that the calculation is 30 x 3
and 30 x 40. But this can be tricky, to
think about Tens. So let’s put a 0 in as a
placeholder straight away (represents
the Ten) and now we just need to think
about 3 x 2 and 3 x 4.”
DIVISION
Short division:
Notes
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MULTIPLICATION
Long multiplication:
Notes
DIVISION
Short division:
Notes
Interpreting remainders as a decimal
Write in the decimal point in the
correct place in the answer.
There are 0 tenths—include this in the
remaining 3, which makes 30.
How many 4’s in 30? There are 7. This
‘7’ is 7 tenths. But 2 tenths remain.
We add this to the 0 hundredths.
How many 4’s in 20? There are 5.
So the answer as a decimal is 120.75
1 2 0.7 5
4 4 8 3 0²0
483 ÷ 4 = 120 r.3
120 ¾
Interpreting remainders as a fraction
3 remain—3 out of 4—this can be represented by a fraction. Can the fraction be simplified?
Interpreting remainders by rounding
eg. 483 sweets are shared equally
between 4 friends. How many sweets
do they get each? (answer = 120)
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Year 6
Curriculum learning objectives
• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
• perform mental calculations, including with mixed operations and large numbers
• use their knowledge of the order of operations to carry out calculations involving the four operations
• solve problems involving addition, subtraction, multiplication and division
• use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.
MULTIPLICATION
Short & Long multiplication:
(see Year 5)
Notes
Areas for application:
Measurements
Geometry
Fractions, decimals, percentages
Ratio and Proportion
Algebra
DIVISION
Short division
Long division:
532 ÷ 14 =
Notes
Areas for application:
Measurements
Geometry
Fractions, decimals, percentages
Ratio and Proportion
Algebra
Like short division, we think about
each place value column:
How many lots of 14 in ‘5’? (no need for
‘500’ as Year 6 children should have a
secure knowledge of place value!)
How many lots of 14 in 53? (=3) This
makes 42—how much of 53 remains?
11…but don’t forget the ‘2 Units’!
How many lots of 14 in 112? (children
can note the times table down the side
to keep track—14, 28, 42…)
There are 8, which makes the answer
38.
Long division using ‘chunking’ can also
be used. See over...
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MULTIPLICATION
Short & Long multiplication:
(see Year 5)
Notes
DIVISION
Long division:
532 ÷ 14 =
Notes
‘Chunking’ - thinking about the number
as a whole rather than the individual
columns / place value.
I know that 10 ‘lots of’ 14 make 140.
Let’s subtract this ‘chunk’ of 140 from
532 = 392.
I know that 20 ‘lots of’ 14 make 280.
Subtract this from 392 = 112.
I know that there are 8 ‘lots of’ 14 in
112.
So...I have calculated that there are
38 ‘lots of’ 14 in 532. Therefore 532 ÷
14 = 38.”
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MULTIPLICATION
Short & Long multiplication:
(see Year 5)
Notes
DIVISION
Long division:
Notes
Interpreting remainders as a decimal
Write in the decimal point in the
correct place in the answer.
There are 0 tenths—include this in the
remaining 12, which makes 120.
How many 15’s in 120? There are 8.
This ‘8’ is 8 tenths. Therefore the
answer is 28.8.
Interpreting remainders as a fraction
12 remain—12 out of 15—this can be
represented by a fraction. Can the
fraction be simplified?