The Vaynor First School
Calculation Policy
December 2008
The long term aim is for our children to be able to select an efficient method of their
own choice asking systematically:
Can I do this in my head?
Can I do this using drawings or jottings?
Do I need to use a pencil and paper procedure?
Do I need a calculation?
This policy was formulated by staff on a Teacher Day in November 2008.
1
Calculation Policy Index
Page Number
Page Number
3
Using and applying objectives YR , 1 ,2
19
Subtraction Year 5
4
Using and applying objectives Y3, 4
20
Division Title Page
5
Addition Title Page
21
Division Foreword
6
Addition Year R
22
Division Year R
7
Addition Year 1
23
Division Year 1
8
Addition Year 2
24
Division Year 2
9
Addition Year 3
25
Division Year 3
10
Addition Year 4
26
Division Year 4
11
Addition Year 5
27
Division Year 5
12
Subtraction Title Page
28
Multiplication Title Page
13
Subtraction Foreword
29
Multiplication Year R
14
Subtraction Year R
30
Multiplication Year 1
15
Subtraction Year 1
31
Multiplication Year 2
16
Subtraction Year 2
32
Multiplication Year 3
17
Subtraction Year 3
33
Multiplication Year 4
18
Subtraction Year 4
34
Multiplication Year 5
2
Using and applying
These objectives are an integral part of every addition lesson.
Year R
•Use developing mathematical ideas and methods to solve practical problems
•Match sets of objects to numerals that represent the number of objects
•Sort objects, making choices and justifying decisions
•Talk about, recognise and recreate simple patterns
•Describe solutions to practical problems, drawing on experience, talking about their own ideas, methods and choices
Year 1
•Solve problems involving counting, adding, subtracting, doubling or halving in the context of numbers, measures or money, for example
to 'pay' and 'give change'
•Describe a puzzle or problem using numbers, practical materials and diagrams; use these to solve the problem and set the solution in
the original context
•Answer a question by selecting and using suitable equipment, and sorting information, shapes or objects; display results using tables
and pictures
•Describe simple patterns and relationships involving numbers or shapes; decide whether examples satisfy given conditions
•Describe ways of solving puzzles and problems, explaining choices and decisions orally or using pictures
Year 2
•Solve problems involving addition, subtraction, multiplication or division in contexts of numbers, measures or pounds and pence
•Identify and record the information or calculation needed to solve a puzzle or problem; carry out the steps or calculations and check the
solution in the context of the problem
•Follow a line of enquiry; answer questions by choosing and using suitable equipment and selecting, organising and presenting
information in lists, tables and simple diagrams
•Describe patterns and relationships involving numbers or shapes, make predictions and test these with examples
•Present solutions to puzzles and problems in an organised way; explain decisions, methods and results in pictorial, spoken or written
form, using mathematical language and number sentences
3
Using and applying
These objectives are an integral part of every addition lesson.
Year 3
•Solve one-step and two-step problems involving numbers, money or measures, including time, choosing and carrying out
appropriate calculations
•Represent the information in a puzzle or problem using numbers, images or diagrams; use these to find a solution and
present it in context, where appropriate using £.p notation or units of measure
•Follow a line of enquiry by deciding what information is important; make and use lists, tables and graphs to organise and
interpret the information
•Identify patterns and relationships involving numbers or shapes, and use these to solve problems
•Describe and explain methods, choices and solutions to puzzles and problems, orally and in writing, using pictures and
diagrams
Year 4
•Solve one-step and two-step problems involving numbers, money or measures, including time; choose and carry out
appropriate calculations, using calculator methods where appropriate
•Represent a puzzle or problem using number sentences, statements or diagrams; use these to solve the problem; present
and interpret the solution in the context of the problem
•Suggest a line of enquiry and the strategy needed to follow it; collect, organise and interpret selected information to find
answers
•Identify and use patterns, relationships and properties of numbers or shapes; investigate a statement involving numbers
and test it with examples
•Report solutions to puzzles and problems, giving explanations and reasoning orally and in writing, using diagrams and
symbols
4
The Vaynor First School
Addition Policy
December 2008
5
Addition - Reception
Begin to relate addition to
combining two groups of objects
Find a total by counting on when 1
object is missing
5
Extend addition to 3 groups
Relate addition of doubles to
counting on
Select two groups of objects to
make a given total of objects
In practical activities and
discussion begin to use the
vocabulary involved in adding and
subtracting
Make 6
2+4
3+3
?
1+5
7 is 5 and 2 more
4+2
Vocabulary
Add, more, and, count on
(from/to), make, sum total,
altogether, score, double, one
more…,how many more to
make..?
?
Examples of Partitioning
Uses a range of strategies for
addition including some mental
recall of number bonds
Uses developing mathematical
ideas to solve practical problems
7
Giant number line
Pupil number tracks
Use of bead strings – including grabbing
Chunks of beads (eg 10 beads)
What is
one more
than 4?
Find one more or one less than a
number from 1 -1 0
6
0+6
7=5+2
5+1
10 = 7 + 3
Adult writes number sentences to match activities
I have 4 red buttons and 2
yellow buttons. So that is 6
buttons altogether. Can you
draw that for me?
It is important to allow opportunities for
Emergent Maths recording as with
Emergent Writing.
More written methods to be introduced in the summer term.
6
Addition – Year 1
Derive and recall all pairs of numbers
with a total of 10 and addition facts for
totals to at least 5; work out the
corresponding subtraction facts.
Extend partitioning of numbers practically as in Reception.
Counting in 10’s from any number, 5’s, 2’s, etc.
Use bead strings – grab chunks and count
Move to how many 10’s/ units
Describe simple patterns and
relationships involving numbers or
shapes
Understand subtraction as ‘take
away’ and find a ‘difference’ by
counting up
Use the vocabulary related to addition
and subtraction and symbols to
describe and record addition and
subtraction number sentences
Use blank numberlines to support
understanding of number bonds.
+1
Relate addition to counting on;
recognise that addition can be done
in any order
Recall the doubles of all numbers to
at least 10
9
+4
10
14
19 20
24
Use straws to extend
understanding of numbers
up to 100
20
e.g. 9 + 5
Secure with number bonds to 10
Then 19 + 5
+1 +4
10
Vocabulary
+, add, more, plus, make, count on
(from/to), sum, total, altogether,
score, double, near double, one
more, two more…, how many more to
make…? How many more..is..than?,
how much more is…?,
23 =2 tens and 3 units
10 + 4 = 20
10+ 10 + 10 + 10 + 10 + 2 = 52
30
Use items e.g. socks to find
doubles to at least 10 by
sorting into two sets and
counting
7
Addition – Year 2
Partition two – digit numbers in
different ways including into multiples
of 10 and 1
Add mentally a one digit number to or
from any two-digit number; use
practical and informal written
methods to add two-digit numbers
Begin with empty numberlines for
partitioning when adding.
Use bead strings to visually support
understanding:
20 + 24 = 44
24 + 20 = 44
Derive and recall all addition facts for
each number to at least 10, all pairs
with totals to 20 and all pairs of
multiples of 10 with totals up to 100.
Use the symbols +, -, x, ÷ and = to
record and interpret number
sentences involving all four
operations; calculate the value of an
unknown in a number sentence (e.g.
□ + 6 = 10, 8 = □ + 5
e.g. 3 + 6 + 4 =
90
Vocabulary
+, add, addition, more, plus, make,
count on (from/to), sum, total,
altogether, score, double, near
double, one more, two more…one
hundred more, how many more to
make…? How many more..is..than?,
how much more is…?,
+ 14 = 104
Partitioning the second number in a sum and
calculate the answer on a blank numberline
+ 30
Use knowledge of number facts and
operations to estimate and check
answers to calculations
Understand that subtraction is the
inverse of addition and vice versa
Use objects to add 3
single digits and support
Mental agility based on
Number bonds to 10
+5
52
82
87
There are 52 people on a bus. 35 more
Get on. How many are there altogether?
Or
52 people were on the bus when it left
the garage. When it arrived in Redditch
there were 87 people on it. How many
people got on during the journey?
52 +
□
= 87
or 87 – 52 = □
Using a number square
to create visualisation
and large numberline to
jump 10 then one
more/less to add near
multiples of 10 e.g. 9
and 11mentally
Use straws to extend
understanding of numbers
up to and beyond 100
100
+ 20
+ 4 = 112
8
Addition – Year 3
Derive and recall all addition facts for
each number up to 20, sums of
multiples of 10 and number pairs that
total 100.
Partitioning
Continue developing the numberline method into a number sentence
using both methods side by side until confident in real life problems.
+ 40
Use knowledge of number operations
and corresponding inverses, including
doubling and halving, to estimate and
check calculations
+7
52
92
99
Add or subtract mentally
combinations of one –digit and twodigit numbers
47 + 52 = 52 + 40 + 7
= 92 + 7
= 99
Develop and use written methods to
record, support or explain addition
and subtraction of two-digit and three
digit- numbers
Adding a near multiple of 10
Using a numberline to model add near
multiples of 10 to two-digit numbers:
28 + 19 is the same as 28 + 20 -1
+ 20
Vocabulary
+, add, addition, more, plus, make,
count on (from/to), sum, total,
altogether, score, double, near
double, one more, two more…one
hundred more, how many more to
make…? How many more..is..than?,
how much more is…?, tens
boundary, hundreds boundary
47
48
38
48
+2
78 80
36 + 12 + 14
or
38 +
47
Children need to be secure adding multiples of
10 to any two-digit number including those that
are not multiples of 10 so 48 + 36 = 84
+30
50 + 12 = 62
76
+9
28
Using known number facts
to mentally calculate several numbers:
-1
28
+ 10
Mrs Dunstan has two part full
boxes of pencils and needs
to know how many she has
altogether. There are 47 in one
box and 52 in another. How many more
does she need for three classes of 28?
86 + 5 = 91
38 + 15
Pencil and paper procedures
In the context of problem solving
76 + 42 = 118
+4
84
9
Addition – Year 4
Use knowledge of addition and
subtraction facts and place value to
derive sums and differences of pairs
of multiples of 10, 100 or 1000
Identify the doubles of two-digit
numbers; use these to calculate
doubles of multiples of 10 and 100
and derive the corresponding halves
Develop and use written records to
record, support and explain
calculations
Add or subtract mentally pairs of two
digit whole numbers (e.g. 47 + 58, 91
– 35)
Refine and use efficient methods to
add and subtract two digit and threedigit whole numbers and £p.
•Vocabulary
•+, add, addition, more, plus,
increase, make, count on (from/to),
sum, total, altogether, score, double,
near double, one more, two
more…one hundred more, how many
more to make…? How many
more..is..than?, how much more
is…?, tens boundary, hundreds
boundary, inverse
Continue using number lines to calculate number sentences, modifying numbers as
appropriate for developmental stage.
Partitioning
Use this for time, money, decimals developing from Y3 work, extending number to 100’s.
A bus journey from Redditch to Cheltenham takes 1 hour 50 minutes
If it left at 8:30 am what time did it arrive in Cheltenham?
1 hr
30 mins 20 mins
total = 1 hr 50
8:30
9:30
10:00 10:20
The bus arrived in Cheltenham at 10:20 am.
Extend confidence in adding near multiples of 10 e.g. 39 / 49
Joe bought two items costing £7.85 and £3.49. How much did he
pay and what change would he get from £15?
+ £3.00
+ 50p
- 1p
£7.85
£10.85
£11.34 £11.35
Joe’s shopping cost £11.34. (Use counting on method for change)
Calculating up to 1,000
Extend work from Year 3 with paper and pencil and numberlines.
+ 100
246
+20
346
+9
366
129 + 246 = 375
375
10
Addition – Year 5
Use knowledge of place value and
addition of two-digit numbers to
derive sums and differences and
doubles of decimals
Use efficient written methods to add
whole numbers and decimals with up
to two places.
Use a calculator to solve problems,
including those involving decimals or
fractions; interpret the display
correctly in the context of
Measurement
Extend mental methods for whole
number calculations
Once totally secure with addition via number lines, move to an expanded method
alongside so children see the link.
The school cook served 358 meals on the first three days of the week and 238 on the last
two days. How many meals did she serve in total?
+200
358
+30
558
+8
588 596
358
+238
200
30
8
596
358
+238
596
1
Most children will swiftly move to the next step which is the compact method
however, is children experience any difficulty they should revert to numberlines /
expanded methods until fully secure.
Move to addition of thousands and decimals using the compact method.
An Olympic runner completes two rounds of the track in the following times:
51.48 secs and 48.72 secs. What was the time of the whole race?
Vocabulary
+, add, addition, more, plus, increase,
make, count on (from/to), sum, total,
altogether, score, double, near
double, one more, two more…one
hundred more, how many more to
make…? How many more..is..than?,
how much more is…?, tens
boundary, hundreds boundary, units
boundary, tenths boundary, inverse
51.48
+ 48.72
100.20
1 1
4687
+ 546
5233
111
11
The Vaynor First School
Subtraction Policy
December 2008
12
Subtraction Policy
Children do not necessarily arrive in school as confident in subtraction as they are in the concepts of addition and
sharing. It depends on the levels of interaction the child has experienced. Those who have been at nursery should be
more confident but some children may enter school with little verbal understanding or conceptualisation beyond singing
rhymes such as 5 fat sausages, ‘s/he’s taken my toy away’ or that if a sweet is taken away from a group there is
something missing.
It is therefore very important that children’s knowledge and understanding in this field is quickly identified and learning is
supported through practical experiences to develop the missing language or concepts that will enable a successful basis
for future work. This may be through role play as a shop worker ‘selling items from the shelf and finding out how many
less s/he has once one has been sold’ or through if ‘I have 5 building bricks and I take one away to build a tower, how
many do I have now?’ or other such activities linked to small / large world play or creative areas. The use of sets of objects
to physically make less and more, taking one..two..three..away or adding will also help see links between these two
concepts. This continues to be important as children use bundles of 10’s or 100’s to develop their understanding of place
value and the effect subtracting these large numbers has.
However, there is another type of subtraction, that linked to using ‘counting on’. This is a concept that few children will
have experienced, although they may be familiar with the term difference, and this is a word that requires use from the
earliest opportunity within school, moving towards practically finding the difference by jumping on numbertracks, counting
up items on numberlines, with lines of little cars – ‘Look you and Jo have different amounts of cars. Lets find how many
less you have than Jo by counting the difference.’
Children need to recognise that whether they find a difference or take away something is getting smaller, this links
through then to division and fractions later.
Recording needs to be pictorially initially, and, as the children move through the school link to the use of numberlines as
the empty number line helps to record or explain the steps in mental subtraction. A calculation like 74 - 27 can be recorded
by counting back 27 from 74 to reach 47.
The empty number line is also a useful way of modelling processes such as bridging through a multiple of ten.
The steps can also be recorded by counting up from the smaller to the larger number to find the difference, for example
by counting up from 27 to 74 in steps totaling 47.
With practice, children will need to record less information and decide whether to count back or forward. It is useful to
ask children whether counting up or back is the more efficient for calculations such as 57 - 12, 86 - 77 or 43 - 28.
13
Subtraction - Reception
Find one more or one less than
any number from 10
Begin to relate addition to
combining two groups and
subtraction to taking away .
Number songs i.e. 5 little ducks, five fat sausages
Rhymes, stories
Scenario’s in role play and outdoor environment
Counting fingers
Physical jumping back and forwards on large number lines and track
In practical activities and
discussion begin to use the
vocabulary involved in adding
and subtracting .
If you start at 5 and jump
back 2 what number do you
land on?
If I have five
fingers, then put
three back down
again how many
do I have left?
Moving to the
language of ‘take
away’
Number beads – counting taking them away
Practical problems – Fruit and snack time / PE/ registers / in the line
Looking at the difference between quantities of objects
Vocabulary
Take (away), leave, count back
(from/to), how many are left/left
over?, how many are gone?, one
less, two less…, how many fewer
is…than…?, difference between,
is the same as, different.
It is important to allow opportunities
for Emergent Maths recording as
with Emergent Writing.
Mia stand on number 1. Chloe stand on number 4.
How many jumps will Mia need to take to be with Chloe?
How many pirates are there?
How many will there be if I take 2
away?
And another 2?
If the parrot jumps from number 2
pirate to number 5 how many jumps
will he make?
Move to using the language of
difference.
1
2
3
4
5
More written methods to be introduced in the summer term.
14
Subtraction - Year One
Compare and order numbers ,using the
related vocabulary; use the equals (=)
sign
Read and write numerals from 0-20 then
beyond; use knowledge of place value to
position thee numbers on a number track
and number line.
Say the number that is 1 more or less
than any given number and 10 more and
10 less for multiples of 10.
Derive and recall all pairs of numbers
with a total of 10 and addition facts for
totals of at least 5 work out the
corresponding subtraction fact.
Continue with the practical experiences from Reception.
Number stories / rhymes and songs.
Using number tracks and lines for counting back.
Take chunks of number
beads to count
on and back.
Partition using bead strings.
Use the vocabulary related to addition
and subtraction and symbols to describe
and record addition and subtraction
number sentences.
Vocabulary
Count back/on (from/to), take away,
difference, subtraction, minus, subtract,
take (away), minus, leave, how many
left/left over?, how many are gone?, one
less, two less…, how many fewer
is…than…?, how much less is…?,
difference between, half, halve, =, equals,
sign, is the same as.
The difference between 8 and 6 is 2.
The difference between 6 and 4 is 2.
What is the difference between 8 and
4?
There are 10 pegs on the coat hanger. I have covered some up.
How many can you see? How many have I hidden?
We can see 6 pegs. There are 4 hidden.
6 + 4 = 10
10 – 4 = 6
-4
Count on and back in ones , twos, fives
and 10’s.
Understand subtraction as take away and
find a difference by counting up. Use
practical and informal written methods to
support the subtraction of a one digit
number from a one digit number or two
digit number and a multiple of 10 from a
two digit number
8 take away 2 equals 6, take
away two more and it equals 4
0 1 2 3 4 5 6 7 8 9 10 Use unit numberlines, moving
to multiples of 10 markings once confident.
Using numberlines marked in multiples of 10, calculate 10 less
starting initially from multiples of 10 moving to any one or two digit number,
bridging through the multiple of 10.
Counting on To form the foundation for using
the counting on method of subtraction children
28 – 10 =
- 2 -8
need to be shown making a difference by
counting on by mentally ‘finding the difference’
by putting the larger number in the head and
0
10
18 20
28 30
counting on as this is more efficient.
Missing numbers within a sentence:
+
= 10
10 =
+
More written methods to be introduced in the summer term.
15
Subtraction - Year 2
Derive and recall all addition and
subtraction facts for each number to
at least 10 all pairs with totals of 20
and all pairs of multiples of 10 with
totals up to 100.
Use knowledge of number facts and
operations to estimate and check
answers to calculations
Begin with empty numberlines for
partitioning when adding.
Use bead strings to visually support
understanding:
28 - 12 = 16
48 = 26
Jane has 18 Christmas Cards. She writes 12
Cards. How many does she has left to send?
- 4
Add or subtract mentally a one digit
number or a multiple of 10 to or from
any two digit number Use practical
and informal written methods to add
and subtract two digit numbers.
Develop informal written methods on the
mental concept of putting one number ‘in
your head’ and counting up to find a
difference. Demonstrate this on a blank
number line, bridging the tens, using bead
strings to support.
Understand that subtraction is the
inverse of addition and vice versa
Use this to derive and record
related addition and subtraction
number sentences.
Jane has 18 Christmas Cards. She writes 12
Cards. How many does she has left to send?
Use the symbols + -- x and = to
record and interpret number
sentences involving all four
operations; calculate the value of
an unknown in a number sentence.
(e.g. □ + 6 = 10, 8 = □ + 5)
Vocabulary
Count back/on (from/to), take away,
difference, subtraction, minus,
subtract, take (away), minus, leave,
how many left/left over?, how many
are gone?, one less, two less…one
hundred less, how many fewer
is…than…?, how much less is…?,
difference between, half, halve, =,
equals, sign, is the same as, tens
boundary,
0
6
-8
8
10
18 20
If I take 12 away from 18 I
have 6 left.
The difference between 12 and
18 is 6 so Jane had 6 cards left.
+6
Which of these methods
is most accurate?
Easiest?
Use number tracks and bead strings and methods highlighted above to support the
subtraction of a multiple of 10 to any 2 digit number, using partitioning.
12
18
20
Mrs Hayward has 44 stickers. She gives 24 to children. How many does she has
left?
? = 44 - 24
-10
20
-10
30
-4
40 44
Subtracting near
multiples of 10 e.g. 9 / 19 18 – 9 = (18 -10) + 1 this should be a mental strategy
backed up by the use of beads
More written methods to be introduced in the summer term.
16
Subtraction - Year 3
Derive and recall all addition and
subtraction facts for each
number to 20 sums and
differences of multiples of 10 and
number pairs that total 100.
Use knowledge of number
operations and corresponding
inverses, including doubling and
halving, to estimate and check
calculations
Add or subtract mentally
combinations of one –digit and
two- digit numbers
Develop and use written
methods to record, support or
explain addition and subtraction
of two-digit and three digitnumbers
Use of support material It is vital that all pupils continue to use the concrete resources from KS1
to support their understanding such as number cards, bead strings, number tracks, bundles of
straws, cubes and counters to consistently support their learning.
Continue to using bridging the nearest multiple, but moving towards combining, or chunking,
numbers to aid a more efficient method.
In Year 3, one class is missing for the day. Mrs Foster-Agg needs to know how many children
are in school. If there are 74 children in the year group, and 27 in the missing class how many
remain in school.
74 – 27 = 47
There are 47 children in school.
Continue to develop the counting on method, as this is particularly useful for
those pupils whose mathematical ability is less sure, linking it to partitioning.
Brian needs to paint the hall walls. He has 28 litres of yellow paint but needs a total
of 57 litres to paint all the walls. How much more paint does he need to buy?
+2
+ 10
+10
+7
Bri
Vocabulary
Count back/on (from/to), take
away, difference, subtraction,
minus, subtract, take (away),
minus, leave, how many left/left
over?, how many are gone?, one
less, two less…one hundred
less, how many fewer
is…than…?, how much less
is…?, difference between, half,
halve, =, equals, sign, is the
same as, tens boundary,
hundreds boundary
28
30
40
50
57
There are 29 more litres of paint
needed to paint the hall.
Next, develop the number line method to subtracting two-digit from three-digit numbers.
Combining of numbers in either counting on or bridging methods should be encouraged.
The year 4 PGL trip cost £128. The money is paid in three parts, the first is £39.
-1
-10
- 28
How much more will need to be paid?
£ 89 more
+ 61
+ 28
needs to
£89£90 £100
be
paid.
£39
£100
£128
Subtracting near
multiples of 10 e.g. 39 / 49
More written methods to be introduced in the summer term.
£128
65 – 39 = (65 -40) + 1 this should be a mental strategy backed up by
the use of beads
17
Subtraction - Year 4
Recognise and continue number
sequences formed by counting on or
counting back in steps of consistent
size.
Use knowledge of addition and
subtraction facts and place value to
derive sums and differences of pairs of
multiples of 10, 100 or 1000
Consolidate the use of methods by the use of one and two step problems involving
Contexts of measure and money, including the use of decimals.
What is the gap between the fastest and slowest laps in a Formula One motor race if the times
Were 22.4 seconds and 17.8 seconds. Which time was the fastest?
22.4 – 17.8 = 4.6 so the gap between
the fastest and slowest is 4.6 seconds.
The fastest time was 17.8 seconds.
Use the knowledge of rounding number
operations and inverses to estimate
and check calculations
Add or subtract mentally pairs of two
digit whole numbers ( e.g. 91-35)
Refine and use efficient written
methods to add and subtract two digit
and three digit whole numbers and £.P
Use a calculator to carry out one-step
and two step calculations involving all
four operations recognise negative
numbers in the display correct
mistaken entries and interpret the
display correctly in the context of
money.
Vocabulary
Count back/on (from/to), take away,
difference, subtraction, minus, subtract,
take (away), minus, leave, how many
left/left over?, how many are gone?, one
less, two less…one hundred less, how
many fewer is…than…?, how much less
is…?, difference between, half, halve, =,
equals, sign, is the same as, tens
boundary, hundreds boundary, decrease,
inverse
Partitioning
74 - 27 = (74 – 20) – 7
= 54 - 7 = 47
This requires children to subtract a single-digit number
or a multiple of 10 from a two-digit number mentally, linked to the number line.
Subtracting a near multiple of 10
Using a numberline to model add near
multiples of 10 to two-digit numbers
which bridge the 100:
135 - 79 is the same as (135 – 80) +1
+1
55 56
- 45
- 35
100
135
More written methods to be introduced in the summer term.
18
Subtraction - Year 5
Use knowledge of addition and
subtraction facts and place value of
two digit numbers to derive sums
and differences and doubles and
halve of decimals
Use the knowledge of rounding
number operations and inverses to
estimate and check calculations
Add or subtract mentally pairs of
two digit whole numbers ( e.g. 9135)
Extend mental methods for whole
number calculation for example to
subtract one-near multiple of 1000
from another (e.g. 6070 – 4097)
Use a calculator to carry out onestep and two step calculations
involving all four operations
recognise negative numbers in the
display correct mistaken entries and
interpret the display correctly in the
context of money.
Vocabulary
Count back/on (from/to), take away,
difference, subtraction, minus,
subtract, take (away), minus, leave,
how many left/left over?, how many
are gone?, one less, two less…one
hundred less, how many fewer
is…than…?, how much less is…?,
difference between, half, halve, =,
equals, sign, is the same as, tens
boundary, hundreds boundary, units
boundary, tenths boundary,
decrease, inverse
Expanded methods
Using counting on method:
Partitioned numbers are then written under one another:
Example: 74 - 27
Expanded method leading to decomposition. This builds on the partition method.:
Example: TU – TU = 74 - 27
60
14
70 + 4
-20 + 7
40 + 7
6
14
74
-27
47
HTU – HTU = 741 -367
700 + 40 + 1
- 300 + 60 + 7
6 13 11
741
-3 6 7
374
More written methods to be introduced in the summer term.
19
The Vaynor First School
Division Policy
December 2008
20
Division Policy
÷ Division starts before school by children being encouraged to ‘Share your sweets/ toys with…’ ‘ Share nicely’
This moves to sharing out one by one things e.g. knives and forks at the table, cakes into party bags etc.
Followed by sorting items – bricks into the box, cars into the garage etc.
÷ Once children arrive in school these techniques should be well established and continue through role play and
classroom activities.
÷ In school it is really important that the ‘sharing’ aspect of division moves quickly to grouping items by criteria
such as size, colour, shape so that children begin to access the language and understanding of grouping
÷ As children move into numbers this all needs repeating so that as items are shared one by one, person by
person, counting follows so 1 for Adam, I for Jo, 1 for Adam, 1 for Jo, 1 for Adam, 1 for Jo becomes – how many
does Jo have? How many does Adam have? Does Jo have more than Adam? Do they have the same / equal
amounts?
÷ Within Reception and Year 1 it is then critical that children move to grouping by amounts – 2 for Jo, 2 for
Adam. There are two left – how can this be shared equally between Jo and Adam? How many will they have
each? etc. Sometimes at this stage there needs to be odd numbers so children understand in this concrete
environment that there can be remainders.
÷ Recording needs to be firstly in the form of drawings and as children move up the school recording on
numberlines as ‘groups/sets of…’. In Year 1 this will be the teacher scribing so that the modelling is carried out in
readiness for Year 2.
÷ Children also need practical experience in Year 1 of looking at a group of items and grouping into sets by the
use of Venn diagrams, or merely by putting a line between groups i.e. teddy bear diagram below where groups
of two have been isolated by a ruler rather than the items being moved. ‘How many
different ways can we make equal groups of teddies.’ In Y2 this can then be
abstracted onto paper.
÷ Children can then confidently understand the concepts behind division and use
their multiples to group on a number line as they move up the school
÷ Finally, if children higher up the school do not understand division, these steps
need to be used again to unpick the gaps in their wall of understanding,
to reinforce the concepts, in order for them to move on.
21
Division - Reception
Observe number relationships
and patterns in the environment
and use these to derive facts.
In practical activities and
discussion begin to use the
vocabulary involved in adding and
subtracting
Sharing milk/ fruit between the class – are there enough?
Using role play area to share knives and forks/ cakes etc equally
Small world play – ie animals in fields
It is important to allow opportunities for
Pairing objects i.e. socks
Emergent Maths recording as with
Emergent Writing.
Count repeated groups of the
same size
Putting balls in bags in PE, items in boxes in tidying up
Share objects into equal groups
and count how many in a group
Sharing children into teams
Know that numbers identify how
many objects are in a set
☺ ☺ ☺ or dinosaurs into like sets, items
☺ ☺ ☺ from a welly walk into groups
Butterfly printing (mirror half)
Use halves of vegetables/ fruits to form pattern – still half no matter what orientation
Vocabulary
Sharing/share/ shared by/into,
grouping/group, lots of, sets,
half/ quarter, equal/equally,
pairs, sorting, counting in
2’s/5’s/10’s, repeated patters,
double, count out/ share out,
left/left over, same as
Socks/items on a line.
Take two away, repeat.
How many are left on the line each time?
How many sets of two?
22
Division – Year 1
Solve practical problems that
involve combining groups of 2,
5, or 10 or sharing into equal
groups.
Recording: draw pictures, writing number sentences, numbers
PE – sorting children into groups, hoops into sets etc.
Science –plants, sorting seeds
Halving – cutting rope /plasticene
Use the vocabulary of halves
and quarters in context
Patterns of shapes:
Say the number that is 1 more
or less than any given number
and 10 more or less for
multiples of 10
Using language of a half, and understanding that it is
Still a half no matter what the orientation
Move from shape to numeral context
Start with number such as 8:
Solve problems involving
halving in the context of
number, measures or money,
for example to ‘pay’ and ‘give
change’
Describe simple patterns and
relationships involving numbers
Vocabulary
Number sentence/sum, number
lines, sharing into, exactly, equal
to, halfway between, divide into
groups, share equally, equal
groups of, left over, operation,
sign
Sorting into groups e.g.
pencils into table groups
What is half of 2?
What is half of 4?
What is half of 8?
4 pots of 3 pencils
3 groups of 4 pencils
12 pencils shared equally
Is 3 in each pot / set
Do lots of examples like this then move
to more abstract examples eg sorting
cubes, small items into hoops +
recording.
What number pattern is there?
What happens if we start at…?
What would we need to do to
get the number above 8?
What is halfway between 3 and 5?
1 and 5?
Repeated subtraction (sharing)
Items on a washing line – take 2 away, place on
floor. How many are left? Repeat. How many pairs/
Sets?
Use practically with bigger numbers and items
eg bricks
I have 20 sweets
If I give you 5 I now have 15, If I give you 5 I now
have 10… etc. etc. How many lots of 5 were there
In 20?
1
3
2
4
23
Division – Year 2
Understand that halving is the
inverse of doubling and derive and
recall doubles of all numbers to 20, and
the corresponding halves.
Derive and recall multiplication facts for
the 2, 5 and 10 times-tables and related
division facts
Represent repeated addition and arrays
as multiplication and sharing and
repeated subtractions (grouping) as
division; use practical and informal
methods and related vocabulary to
support division, including calculations
with reminders
Use the symbol +, -, x, ÷ and = to
record and interpret number sentences
involving all 4 operations; calculate the
value of an unknown in a number
sentence.
Continue practical division from Year 1 with
practical grouping and sharing, including
remainders e.g.
Take sets of beads
and demonstrate
repeated sets of
numbers being
moved for division.
Use of blank number lines
There are five apples in a bag.
Jaswinder has 15 apples.
How many bags can she make?
1 bag of 5
Mrs Roberts is getting ready for sports day.
She shares 17 balls into three buckets.
How many were in each bucket?
0
2 bags of 5
5
10
3 bags of 5
15
Jaswinder can make 3 bags of 5
apples.
There are 5 balls in each bucket with 2 left over.
15 ÷ 3 = 5 r 2
Remainders
Mum has 70p in 10p coins to
share equally between her three
children. How much will each
child have? Has Mum any left?
I lot of 20p 2 lots of 20p 3 lots of 20p with 10p left
Use knowledge of number facts and
operations to estimate and check
answers to calculations (inverse)
E.g. □ x 2 = 10, 2 = 10 ÷ □
10p 20p 30p 40p 50p 60p 70p
Each child has 20p and Mum has 10p left
Vocabulary
Array, rows and columns, equal groups
of, divide, divided by/into, left over,
calculation/calculate, symbol, one
half/quarter, exact/exactly, fraction.
Repeated addition / subtraction
15 = 5 + 5 + 5 so 15 ÷ 5 = 3
15 = 3 + 3 + 3 + 3 + 3 so 15 ÷3 = 5
Grouping, particularly of large numbers
Using Cuisenaire / bundles of straws etc. physically
Group ‘sets of’ to find the division of large numbers and
link to number lines.
Resource: Grouping ITP to demonstrate as it has an
effective, interactive picture which links to grouping on
a number line.
Fractions: finding ½ or ¼ of a shape or number24
Division – Year 3
Understand that division is the inverse
of multiplication and vice versa; use
this to derive and record related
multiplication and division facts
Use practical and informal written
methods to divide two-digit numbers
e.g. 50 ÷ 4; round remainders up or
down depending on the context
Find unit fractions of numbers and
quantities (e.g. ½ 1/3 ¼ and 1/8 of 12
litres)
Derive and recall multiplication facts
for the 2,3,4,5,6, and 10 times-tables
and the corresponding division facts
Use knowledge of number operations
and corresponding inverses, including
doubling and halving, to estimate and
check calculations
Read and write proper fractions (e.g.
3/7. 9/10) interpreting the denominator
as the parts of a whole and the
numerator as the number of parts;
identify and estimate fractions of
shapes; use diagrams to compare
fractions and establish equivalents
Vocabulary
Remainder, method, 1/3 and 1/10
(third/tenth), division, round up/down,
numerator/denominator
Division known as repeated subtraction
or addition
A baker bakes 24 buns.
She put 4 buns in every
box. How many boxes can
she fill?
Use of place value boards and digits to ÷ by 10
H
T
U
Remainders – round up / down
Farmer Brown has 26 eggs. He packs them in
boxes of 6. How many boxes will he need?
I box of 6
2 boxes of 6 3 boxes of 6 4 boxes of 6 r. 2
Using the number facts:
6 x 4 = 24 24 ÷ 4 = 6
●●●●
●●●●
●●●●
●●●●
0
6
12
18
24 26
Farmer Brown has 4 full boxes with 3 eggs left over.
He needs 5 boxes for all the eggs.
I have got £27. Tickets cost £5 each. How many
tickets can I buy? How much change will I have?
Work from known.
So 6 x 2 = 12
Double 12 is the same as
6 x 4 = 24
The baker fills 6 boxes
I ticket
£0
£5
2 tickets 3 tickets 4 tickets 5 tickets
£10
£15
£20
5r2
£25 £27
5 lots of £5 with £2 left over
5 tickets cost £25 with £2 remainder.
Fractions
Explore practically using actual items
E.g. 3 chocolate bars shared between 7 people. Each get 3/7 of a chocolate
Bar. Repeat with other fractions, which is a greater fraction?
Use other concrete examples then move to using paper and pen to record findings
Using labelled drawings / diagrams.
25
Division – Year 4
Derive and recall multiplication facts
up to 10 x 10, the corresponding
division facts and multiples to 10 up
to the tenth multiple
Extend the use of numberlines
to chunking of multiples up to 10 x multiples, moving
To small remainders requiring rounding up/down
There are 42 children in a class.
The teacher puts the children into teams of four.
How many full teams are there?
Multiply and divide numbers to
1000 by 10 and then 100 (whole number
answers) understanding the effect; relate to
scaling up or down.
Identify the doubles of two-digit numbers; use
these to calculate doubles of multiples of 10
and 100 and derive the corresponding halves
Develop and use written numbers to record
support and explain division of two-digit
numbers by a one-digit number, including with
remainders e.g. 98 ÷ 6
Use a calculator to carry out one step and two
step calculations involving all four operations;
correct mistaken entries and interpret the
display correctly in the context of money
Find fractions of numbers, quantities or
shapes (e.g. 1/6 of 30 plums)
Use diagrams to identify equivalent fractions
(e.g. 7/8 ¾, or 70/100 and 7/10); interpret
mixed numbers and position them on a
number line e.g. 3 ½ )
Recognise the equivalence between decimal
and fraction forms of one half, quarters, tenths
and hundredths
Vocabulary
Equal proportions, divisible by, factor, decimal
fractions, inverse, quotient, equivalent,
decimal point/ place
10 x 4
Use of place value boards and digits to
show ÷10 and ÷ 100
Th
H
T
U
1/10
1/100
+2
0
40 42
42 = (10 x 4) + 2
There are 10 full teams.
Move to chunking where the amounts are greater than multiples of 10, estimate first:
It takes 756 days to write a new X Box 360 game. How many weeks is this?
10 x 7
+
8 x7
108 weeks
0
700
756
Moving to rounding up / down estimate first: 63 children are going camping.
Each tent sleeps 4 children. How many tents are needed?
Activate prior learning practically
10 x 4
+
5 x4
+3
i.e. division of a cake, what
Information do children know?
What number sentences can children
devise?
0
40
60
63
Give 2 bars of chocolate per
(10 x 4) + (5 x 4) = 15 x 4 r 3
table with instructions.
16 tents are needed because 3 children still need somewhere to sleep.
I.e. find 2/7, 3/9 then order
find biggest, smallest, equate
Move to: expressing remainders as quotients, estimating first
to other equivalent fractions.
Sophie prepares 198 pizzas to share between 4 year groups. How much
will each year group receive? Express your answer as a quotient/ fraction.
10 x 4 10 x 4 10 x 4 10 x 4 9 x 4 r 2
0
40
80
120
160
196 198
196 ÷ 4 = 49 r 2/4 so, Sophie gives each year group 49 ½ pizzas.
26
Division – Year 5
Refine and use efficient written
methods to multiply and divide
HTU x U, TU x TU, U.t X U and
HTU ÷ U
'Short' division of TU ÷ U can be introduced as a
more compact recording of the mental method of
partitioning, linking back to number line work.
Find fractions using division
(e.g. 1/100 of 5 kg), and
percentages of numbers and
quantities (e.g. 10%, 5 % and
15% of 80)
Short division of two-digit number can be introduced
to children who are confident with multiplication and
division facts and with subtracting multiples of 10
mentally, and whose understanding of partitioning
and place value is sound.
Use knowledge of rounding,
place value number facts and
inverse operations to estimate
and check calculations.
First, children use the expanded notation within the
brackets, moving to the shortened version with a
carry digit which represents the two tens.
Use understanding of place
value to multiply and divide
whole numbers and decimals
by 10, 100 or 1000
Once the principle is grasped children can move to
‘How many nines in 90?’ or ‘What is 90 divided by 9?’
Recorded in the beginning of a chunking method and
showing a remainder.
3.
Finally, children should quickly move to HTU ÷ U
using a chinking method, linking it back to a number
line initially to support understanding.
4.
Identify pairs of factors of twodigit whole numbers and find
common multiples
Recall quickly multiplication
facts up to 10 x 10 and use
them to multiply pairs of
multiples of 10 and 100; derive
quickly corresponding division
facts
Vocabulary
Proper / improper fractions,
divisibility, percentage
1.
2.
27
The Vaynor First School
Multiplication Policy
December 2008
28
Multiplication - Reception
Count aloud in ones, twos,
fives or tens.
Songs, rhymes, number lines, counting stick, use of coins
Count repeated steps of the
same size.
Jumping in 2’s on a number track
It is important to allow opportunities for
Emergent Maths recording as with
Emergent Writing.
Counting in 10’s on a bead string
●●●●●●●●●●○○○○○○○○○○
Using items around the classroom and labelling
Draw around hands and label
Vocabulary
Compare, double, half/halve,
pair, count out, share out,
left out, groups of, lots of
How many socks? How many pairs of socks?
1
2
3
4
5
29
Multiplication – Year 1
Say the number that is 10 more
for multiples of 10.
Count on in two’s, fives and
tens and use this knowledge to
derive the multiples of 2, 5 and
10 to the tenth multiple.
Recall the doubles of all
numbers to at least 10.
Use the vocabulary related to
addition/ subtractions and
symbols to describe and record
addition/subtraction number
sentences
Solve practical problems that
involve combining groups of 2,
5 or 10 or sharing into equal
groups.
Vocabulary
Double, near double, count
on/up/ back, how many times?,
pattern, pair, every other, count
in ones…2’s…3’s…10’2..etc,
number sentence, total,
altogether, sum (when about
addition), equal groups of, count
2…5…10.. more etc, sign
Counting
Counting in 2’s e.g. socks, shoes, animals, legs…
Counting in 5’s e.g. fingers, fingers in gloves, toes….
Counting in 10’s e.g. fingers, toes…
Looking at rows
3+3
2 groups of 3
Looking at columns
2+2+2
3 groups of 2
Counting in 2p’s, 5p’s, 10p’s
Using concrete items to support early
multiplication:
3 pairs of socks.
How many socks?
Jo had 4 party bags of 2 biscuits for his friends.
How many biscuits did he use?
VAK resources
Number songs, counting sticks, number
tracks, number squares
Doubling 2 digit numbers to 20
Picture cards (sets of objects)
Pegboards ○ ○
show arrays
Use bead strings
Pictures / marks
There are 3 sweets in one bag.
How many sweets are there in
5 bags?
○ ○ to
1 group of 4 2 groups of 4
0
4
8
Double 4 is 4 + 4 = 8
30
Multiplication – Year 2
Understand that halving is the inverse
of doubling and derive and recall
doubles of all numbers to 20.
Derive and recall multiplication facts
for the 2, 5 and 10 times-tables;
recognise multiples of 2, 5 and 10.
Represent repeated addition and
arrays as multiplication; use practical
and informal methods and related
vocabulary to support multiplication.
Use the symbol +, -, x, ÷ and = to
record and interpret number
sentences involving all 4 operations;
calculate the value of an unknown in
a number sentence.
Use knowledge of number facts and
operations to estimate and check
answers to calculations (inverse)
E.g.
□ x 2 = 10,
2 = 10 ÷
□
Rapid recall
of number
doubles.
Missing numbers
Doubling
multiples
of 5 up to 50
□ x 2 = 14
□ x = 14
7 x □ = 14
□=2x7
14 = □ x 7
14 = 2 x □
14 = x □
Arrays to support understanding
Use objects, peg
boards and counters,
alongside squared
paper, to create a visual picture
of arrays then move to numberlines.
An additional resource is the spreadsheet on the
Framework called array creator. This helps children
see the link between division and illustrate inverse.
●●●●
●●●●
●●
●●
●●
●●
4 x 2 or 4 ÷ 2
2 x 4 or 2 + 2 + 2 + 2
Vocabulary
groups of, lots of, sets of, times,
multiplied, double, multiple of,
place, place value, x, times,
multiply, multiplied by, multiple of
once, twice, three times, four
times, five times... ten times...
times as (big, long, wide and so
on) repeated addition, array row,
column, inverse operation, sign
7x2=□
Language of multiplication
+4
+2
0
1
+4
+2
2
3
+2
4
5
+5
+5
0
+5
5
+5
10
+5
15
20
+5
= 30
25
30
What multiplication
5 x 6 = 30
sentences could
5 multiplied by 6
we write about
6 groups of 5
this?
6 hoops of 5
6 classes each have 5 house points.
How many house points are added to
their house on Monday?
Begin to partition
First using informal methods
19
20
+
18
= 38
Followed by supported partitioning
3 x 10
3x5
+2
6
7
8
0
30
45
3 x 15 = 3 x 10 + 3 x 5
Cut out squared paper to represent arrays.
= 30 + 15 = 45
Solve problems using all types of arrays.
Using money to solve multiplication problems e.g Miss Parry bought 3 oranges for 15p each.
How much did she spend?
31
Multiplication – Year 3
Derive and recall multiplication facts for
the 2,3,4,5,6, and 10 times-tables and
the corresponding division facts;
recognise multiples of 2,5 or 10 up to
100.
Informal Mental Methods
12
Use knowledge of number operations
and corresponding inverses, including
doubling and halving to estimate and
check calculations
Vocabulary
groups of, lots of, sets of, times,
multiplied, double, multiple of, place,
place value, x, times, multiply,
multiplied by, multiple of
once, twice, three times, four times,
five times... ten times...
times as, repeated addition,
array row, column, inverse, product,
remainder, partition
4
X
Reciting tables
Leapfrog summer
term
3
Use of place value boards and digits
Th
Differences of multiples of 10 and
numbers of pairs that total 100
Understand that division is the inverse
of multiplication and vice versa; use this
to derive and record related
multiplication and division number
sentences.
÷
÷
2x6
Multiply one-digit and two-digit numbers
by 10 or 100 and describe the effect.
Use practical and informal written
methods to multiply two-digit numbers
e.g. 13 x3; rounding remainders up or
down depending on the context
12
4x3
3x4
6x2
H
T
Place value
grids to
show x 10,
x 100
U
Informal written methods
5 hops / jumps of 2
1x2
2x2
3x2
4x2
5x2
2x4
I have
12
stickers
20 + 10 = 30
Move to grid method using
answers under 100
X
10
5
20
10
Use a rectangular array to
show multiplication by 10.
e.g. 6 x 10 =
.
3x4
The egg method
Partitioning
Doubling: 10 + 5
2
●●●●●●●● 8 x 4 =32
●●●●●●●● 4 x 8 = 32
●●●●●●●● 32 ÷4 = 8
●●●●●●●● 32 ÷8 = 4
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
2 x 18 = 36 36 ÷ 2 = 18
18 x 2 = 36 36 ÷18 = 2
I have 4 stickers. If you had 3 times as
many stickers, how many would you
have?
1x4
Arrays and repeated
addition
Understand multiplication as
repeated addition and
continue to use arrays
for x and ÷
36
Number sentences with
missing numbers
e.g
60
+
12 = 72
□ x 2 = 20
40 = 5 x □
= 30
32
Multiplication – Year 4
Derive and recall multiplication
facts up to 10 x 10, the
corresponding division facts and
multiples to 10 up to the tenth
multiple.
Multiply and divide numbers to
1000 by 10 and then 100 (whole
number answers), understanding
the effect; relating to scaling up or
down
Identify the doubles of two-digit
numbers; use these to calculate
doubles of multiples of 10 and 100
and derive the corresponding
halves
Develop and use written methods
to support and explain
multiplication and division of twodigit numbers by a none digit
number
Vocabulary
groups of, lots of, times,
multiplication, double, place value,
x, times, multiply, inverse,
multiplied by, multiple of
once, twice, three times, four
times, five times... ten times...
times as, repeated addition,
array row, column, inverse,
product, remainder, partition
Use of place value boards and digits to show x 10 and x 100
Th
H
T
U
1/10
1/100
Partition
Continue to use arrays:
Informal written methods
Continue to use partitioning with numberlines
New books are packed boxes of 7. If Mrs
Dunstan orders 38 packs how many books
will she have?
30 x 7
8x7
0
210
18 x 9 = 162
18 x 9 = (10 x 9) + (8 x 9) = 162
266
Mrs Dunstan will have 266 books.
Extend the grid method to answers
that break the 100’s barrier
X
30
8
7
210
56
= 161
33
Multiplication – Year 5
Extend mental methods for whole
number calculations, for example
to multiply a two-digit by one-digit
number (e.g. 12x9, to multiply by
25 (e.g. 16x25
Refine and use efficient written
methods to multiply and divide
HTU x U, TU x TU, U.t x U and
HTU ÷ U
Recall quickly multiplication facts
up to 10 x 10 and use them to
multiply pairs of multiples of 10
and 100; derive quickly
corresponding division facts
Identify pairs of factors of two-digit
whole numbers and find common
multiples (e.g. for 6 and 9)
Use knowledge of rounding, place
value, number fats and inverse
operations to estimate and check
calculations.
Vocabulary
As year 4
Use of place value boards and digits to show
x 10 and x 100 x 1000
Th
H
T
U
1/10
1/100
1/1000
Informal written methods move to two- digit numbers
x two-digit numbers
X
30
2
70
2
2100
60
140
4
2160
44
2304
2. If children are ready, to move to a
compact method of multiplication,
using a carrying figure below the line.
If, after practice, children cannot use the
compact method without making
errors, they should return to the
expanded method as previous.
38 The step here involved adding 210
X 7 and 50 mentally with only the 5 in the
266 recorded. This highlights the need for
5
children to be able to add a multiple
of 10 to a two-digit or three-digit
number mentally before they reach
this stage.
Partitioning
276
400 140
12 = 552
Linking informal and formal multiplication
1. The next step is to link numberline and grid
methods with an expanded column format,
but showing the working. Children should
describe what they do by referring to the actual
values of the digits in the columns. For example,
the first step in 38 × 7 is ‘thirty multiplied by
seven’, not ‘three times seven’, although the
relationship 3 × 7 should be stressed.
30 + 8
´ 7
210
56
266
30 ´ 7 = 210
8 ´ 7 = 56
34