Woodlands
Primary School
Mathematics Calculation Policy
Woodlands Primary School
Guidance for the Calculation Policy
Woodlands Primary School: Calculation policy
Policy reflects: concrete (do it) abstract (see it!) visual (remember it!) communication (record it!)
At Woodlands, we believe that an ability to calculate mentally lies at the heart of calculation.
The foundations of good number sense, mental calculation and the recall of number facts need to be established thoroughly before formal written methods are introduced.
National Curriculum Core Aims:
1. Developing Fluency
 Regular opportunities will be provided so that children become fluent in their ability to apply their knowledge with increasing accuracy and speed to a range of problems.
2. Reason Mathematically
 All children will have regular opportunities to follow lines of enquiry, conjecture relationships and make generalisations.
 Use of specific mathematical vocabulary will be promoted and modelled by adults regularly so that children become skilled in ‘developing an argument, justification or proof using mathematical
language.’
3. Problem Solving
 Throughout all stages of the policy, we aim to give pupils the opportunity to acquire a range of strategies which can be applied to solve problems, particularly more complex problems where two or
more steps are needed.
 Opportunities to solve different types of problems will be provided regularly so that children are able to apply their mathematics with ‘increasing sophistication.’
 Through regular exposure to problem solving, children will become skilled at breaking complex problems down into simpler steps; they will tackle problem solving with a high level of independence,
making decisions and showing resilience as they persevere in finding a solution.
Progression from mental to written methods:
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Our aim is for all children to have a reliable and efficient method for calculating in all four operations.
The four operations will be known as addition, subtraction, multiplication and division from Foundation stage to Year 6.
Mental strategies will include the teaching of informal recording and personal jottings.
November 2016
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We will use models and images to support the development of mathematical concepts. Where particular resources need to be used, they have been identified in the policy.
We will use concrete models, abstract images and visualising to secure understanding and build memory.
The calculation policy tracks the mental skills required for each written method and these should be taught progressively.
The policy will also support the diagnosis of errors and can be used to make judgements about the stage of development.
Specific language (blue text) which can be used to support and explain calculation has been notated with a blue speech bubble.
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It is expected that all staff and pupils will use this language.
Use of the Calculator
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The calculator will continue to be used as a teaching and learning tool across school. It should not replace written calculation methods but may be used in lessons when the focus is on other skills
e.g. problem solving.
Decision Making
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Children should have sufficient understanding of the calculation in front of them so that they can decide which method to use – mental, mental with jottings, informal written method,
formal written method or a combination of these methods. Our aim is for children to apply these accordingly, with confidence.
Editing
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Children should be actively encouraged to check and edit their work. When work is edited, this should be completed using red edit pen (as stated in the school presentation policy).
Cross curricular links
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Cross curricular links are planned for to ensure children experience mathematics in a wider range of contexts. As a school, we believe it is vital that children gain an understanding of how
mathematics is linked to real life contexts.
Implementation and review
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Staff will receive the updated policy – November 2016.
Staff will use the policy to plan for progression in calculation.
Implementation of the policy will be reviewed regularly by the Mathematics subject leader.
Bar Model Calculation Appendix to be used alongside calculation policy – November 2016
November 2016
Woodlands Primary School: Progression in Number Sense
Concrete: Abstract: Visual
Number sense: nrich.maths.org :: Mathematics Enrichment :: Number Sense Series:
Developing Early Number Sense
nrich.maths.org :: Mathematics Enrichment :: Number Sense Series: A Sense of 'ten' and
Place Value
Learning to count with understanding is a crucial number skill, but other skills, such as
perceiving subgroups, need to develop alongside counting to provide a firm foundation for
number sense.
To begin with, early number activities are best done with moveable objects such as counters,
blocks and small toys.
Models and images
By simply presenting objects (such as stamps on a flashcard) in various arrangements,
different mental strategies can be prompted.
After the essential experiences of practical apparatus more static materials such as 'dot
cards' become very useful.
Abstract – Counting ITP
Which are instantly recognised and which are counted?
If mental strategies such as these are to be encouraged (and just counting discouraged) then
an element of speed is necessary. Seeing the objects for only a few seconds challenges the
mind to find strategies other than counting. It is also important to have children reflect on
and share their strategies.
Combining – Abstract – Counting ITP
Concrete models
Random arrangements that can be counted.
Random arrangements that can be ‘subitised’ e.g. 5 beads and 2 beads.
Linear arrangements, such as a bead string
Recognising the 4 and then the 3, combining this to make 7.Counting on from 4
rather than counting all 7
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Learning focus
Distinguish between quantities; recognise
when a group of objects is more than one
Concrete
Abstract
Use published pictures and photographs of every day objects
recognisable to the children (both inside and outside of the
classroom) that show only one object and more than one.
For example: fruit, animals, cars
Provide opportunities through play for children to identify sets containing
‘only one’ object and those containing more than one.
One bear all alone
You will need: small plastic bears or other objects, opaque plastic cups or
bowls
Place three or four bowls upside down on a table. Place a single bear
under one of the bowls and different quantities of bears under the
others.
Invite children to help you. Tell the children that you are looking for the
lonely bear, the one all on its own. Ask the children to take turns to turn
over a cup or dish, encouraging them to describe what they have found.
Use prompts and questions such as: Tell us what you have found. Have
you found one bear or more than one?
Encourage children to take the bears that they find and to place them in
a line in front of them. Prompt children to compare the number of bears
that they have, by asking questions such as: Do you think that Max or Evie
found more bears? How could we check?
Invite the children to play again by placing their bears back under a cup.
The child who found the lonely bear could mix the cups around before
you play again.
Taking ideas further: Provide appealing objects and containers for
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children to use in their play. Ensure that you provide some containers with
lids to encourage children to guess which or how many objects may be
inside.
Recognise groups with one, two or three
objects
Subitising with objects.
For example, provide equipment that has to be counted e.g. matches,
multilink and counters. Children then count out 1, 2 and 3 from each set
of objects.
Any activity that uses images of one, two or three objects.
For example:
One, two or three?
You will need: three hoops, number cards 1, 2 and 3, some
photographs of objects normally found individually or in
pairs, images of groups of three objects, a camera
Place a couple of cards or objects into each hoop so that one
hoop contains individual objects (such as a photograph of a
nose, a bin), a second contains objects normally found in twos
(such as a photo of two eyes, a pair of socks) and the third
contains images of objects in groups of three (such as wheels
on a tricycle, legs on a three-legged stool, triplets).
Invite children to help you to add to the display. Encourage
them to look at the objects in each hoop so far and to suggest
how you have sorted the objects. If no one suggests that you
have sorted them by number, show children the number cards
1, 2 and 3 and say that these cards could be used to label the
hoops. Which number goes with which hoop? Why?
Encourage children to go for a walk (in pairs or a small
group), inside and outside, to find other items that could be
added to the display. Explain that if children cannot bring the
actual objects, they can make a drawing or use the camera to
take a photograph. Walk around with children, observing if
they can instantly identify whether sets contain one, two or
three items.
Taking ideas further: Play ‘Odd one out’, asking children to
close their eyes while you move an object from one of the
hoops to another. Can the children identify which object you
have moved?
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Match and compare the number of
objects in two sets, recognising when the
set contains the same number of objects.
Provide opportunities for children to compare two sets of objects through
malleable play.
(page 9 of enabling environments)
Abstract – Counting ITP
Use counting ITP to compare 2 sets of objects.
Find five
You will need: hoops or plates, five each of three interesting objects (to
hide)
Show children objects that you have chosen to hide and explain that there
are five of each of these hidden around the building, inside and outside.
Invite children to go on a hunt to find the hidden objects. Suggest that as
soon as they find an object, they bring it back and place it on the table or
floor.
Encourage children to hunt in pairs so that they can discuss their ideas.
Observe and listen to children as they hunt. Discuss the hunt with
children, for example: I can see that you have found two key-rings. Have
you found anything else? After a while, call children together by the pile
of objects to look at what has been found so far. Lead a discussion about
what has been found. Use prompts such as:
• Could we organise the objects to see clearly what we have found so far?
(Use hoops or plates to sort the objects.)
• How many stars have we found?
• Can you show me this number on your fingers?
• Have we found all of them yet?
• Which object have we found least of (or most of) so far? How do you
know?
Encourage children to hunt for the remaining objects until they are all
found.
Taking ideas further: Change the number of each object hidden. Ask
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children if they can work out how many of each object they still need to
find.
Instantly recognise without counting
familiar patterns of up to six objects.
Use apparatus such as dice, playing cards and dominoes which have a
structured pattern for instantly recognising a number and get children to
replicate using counters.
Use pelmanism memory game to match familiar patterns of
up to six objects.
Make dominoes
You will need: sticky dots, rectangular cards in different colours, cut into
domino-shaped cards with a central line
Invite children to make their own sets of dominoes with up to four dots in
total. Encourage children to explain what dominoes look like. Give children
different-coloured card.
Explain that you are going to start by making all of the different
dominoes that you can that use only one dot altogether. Ask everyone to
have a go at making a domino that has only one dot.Make a domino
November 2016
yourself so that you can model the use of mathematical language, for
example:I have one dot. I am going to place it on this side of my domino.
Compare children’s examples.Show that it doesn’t matter which side you
stick the dot by turning the dominoes around.
Next make dominoes that have two dots. Encourage children to discuss
what they are doing and to compare their dominoes. Use prompts such as:
• You have taken two sticky dots. Where are you going to stick them now?
• Paul and Sam’s dominoes look different. Why? Do they both have two
dots altogether?
Encourage children to make sure that they have all possible ways of using
the dots each time. Where necessary, use questions such as: Sam, you
have made a two-dot domino with both dots on the same half of the
domino. How could you put two dots onto this domino so that they are not
on the same half?
Continue until children have made all possibilities up to a total of four
dots. Encourage them to look at the sets they have created. Ask questions
such as:
• Can you find all of the dominoes with three dots on one half?
• Who can describe the way that the dots are organised on this domino?
• How many dots altogether are there on this domino?
Taking ideas further: Children could play games with their sets of
dominoes.
Instantly recognise, without counting,
organised and random arrangements of
small numbers of objects
Practical Subitising with numbers greater than 6, but within the counting
experience of the child.
Bead strings, coins, counters, straws, matchsticks can all be used to set up
arrangements using the familiar patterns previously explored (using dice,
dominoes and playing cards), linear arrangements (beads) as well as
random arrangements.
Abstract – Counting ITP
Opportunities for children to recognise random and organised
arrangements.
November 2016
Going dotty
You will need: 3 × 3 squared paper (to use to make dot patterns for
numbers), sticky dots to fit into the squares, a dotty dice
Invite children to help you make cards to play a game. Ask children to
show you the face of the dice that has six dots on it. Encourage children
to consider the pattern of the dots. Use prompts such as: How did you
recognise so quickly that this face has six dots? What is special about the
pattern? Respond to comments, reinforcing mathematical vocabulary, for
example: Jamie said that the dots are in two lines. How many dots are in
each line? The six dots are in two rows (pointing), with three dots in each:
one, two, three... one, two, three. Ask each child to take a piece of
squared paper and some sticky dots and make the same pattern of six
dots. Encourage them to talk about what they are doing. Use prompts such
as: How many dots have you stuck down so far? What are you going to
do next?
Show children a different arrangement of six dots and ask them how
many dots they can see. Establish that there will be lots of ways to
organise six dots onto squared paper. Ask each child to try to find a
different arrangement.
Work together in a similar way to produce different patterns of five,
four and three dots and use them in a game. Place the patterns in the
middle of the table, face down. Then children take turns to select two
patterns. If they show the same number of dots then they keep the pair.
Taking ideas further: Use the cards for similar matching games such as
snap.
Use the skills of subitising to combine two
sets of objects with or without counting
in the range of 1-10.
Work with two sets of objects that children arrange into recognisable
formations. Are children able to say how many there are altogether?
Random to recognisable patterns for addition.
November 2016
Combining boxes
You will need: boxes containing different numbers of items (each labelled
with the number of items inside), number cards to 20 or blank labels
Pick two boxes and show them to the children. Ask what they think the
number on each box tells them. Check by asking children to count the
objects in each box.
Explain: We are going to put the objects from these two boxes together.
Abstract – Counting ITP
We need to work out how many objects that will give us altogether. How
could we do that? Encourage children to suggest different methods and
try them. Include counting on methods, for example: There are eight
objects in this box and three in the other. Let’s add the three objects into
this box. Ask children to count aloud as each object is dropped in, giving
the total number of objects in the box so far: We started with eight,
that’s nine, ten, eleven. How many objects are there altogether? Write the
total on a sticker.
Ask children to work in pairs, to choose two boxes and combine the objects
into one box, working out the total to record on the box. Observe how
children go about this.
Taking ideas further: Carry out similar subtraction activities, taking some
items from one box to put into another and working out how many objects
are left in the first box.
November 2016
Addition
Counting objects, partitioning and
recombining sets using practical
apparatus.
Understand that the number gets
bigger.
Addition is commutative.
Use number tracks to develop
counting skills, forwards and
backwards.
Subtraction
Know that the number gets smaller
because objects have been removed
from the set (unless you subtract
zero).
Stage A (R/Y1)
Multiplication
Jumping along number lines in jumps of 1,2,5
& 10.
Counting on and back in steps of 1,2 and 10.
Sharing objects equally, practical contexts.
Repeated addition, practical demonstrations.
Models and Images charts
Pictorial recording.
Practical models of subtraction
Counting back on fingers, orally,
number lines (Take-away).
Find the mathematical difference
(Counting on).
Models and Images charts.
Doubles and grouping. Grouping is a random
arrangement of a quantity into equal groups.
Grouping, in practical contexts.
GROUPING ITP
Arrays are a rectangular arrangement to
show the equal groups.
COUNTING ITP
Pictorial recording of practical
experiences.
Teacher modelling of number
sentences and addition as
commutative.
Division
Use cross curricular links (PE) and
purposeful objects such as sock and shoes/
animals in the ark to get into groups.
Sharing models such sharing an apple or a
Satsuma.
How many cars can you make if you have 8
wheels?
This is an array
Grouping and sharing, practically. (NB If
the answer is in the same units as the
dividend, it is sharing. If the answer is in
different units, it is grouping.)
November 2016
Example:
Dividend ÷ Divisor = Quotient
£20 ÷ 4
= £5 (Sharing)
£20 ÷ £4
= 5 (Grouping)
Use of arrays to show that multiplication is
commutative. Changing the order does not
affect the answer. Peg boards are a useful
model.
NUMBER FACTS ITP
Practical demonstrations of take
away.
Use the language of ‘lots of’, ‘groups of’ and
‘sets of’ for ‘x’.
There were 9 balloons. Two popped.
How many are left?
Once numbers can be written,
number sentences can be recorded.
9–2
=7
Modelling of commutative layout.
November 2016
Bar Model
To have experience of ‘=’ sign as
last stage in calculation.
GROUPING ITP
ADDITION AND SUBTRACTION
EXCEL
Pictures to show 2 lots of 3 or 3 lots of 2.
10 – 6
=4
Find the mathematical difference
where numbers are close together.
“How many more do I add to 7 to
get to 9?”
DIFFERENCE ITP
For Subtraction, reinforce the order
of calculation. E.g BIGGEST
NUMBER FIRST
9–7
=2
______________
7 8
9
Vertical number line to show the
mathematical difference.
9
8
7
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Link to Diennes model
e.g.
9 - 7= 2
Link to Beadstring model (for
difference).
Addition
Subtraction
Key skills of knowing number bonds
to 10.
Subtraction sentences and jumps
along number lines. (backwards for
take away – left and forwards for
difference – right) along number
lines.
Develop knowledge of fact families,
e.g. 2, 5, 7.
EXCEL ADDITION AND
SUBTRACTION TRIOS
EXCEL ADDITION AND
SUBTRACTION TRIOS
Stage B (Y1/2)
Multiplication
Pictorial repeated addition. Grouping is a
random arrangement of a quantity into equal
groups.
Arrays are a rectangular arrangement to
show the equal groups.
Division
With practical equipment:
Counting on and back in 2s, 5s and 10s and
begin counting in 3s and 4s.
Grouping as repeated addition along the
number line.
Introduce the ÷ symbol once repeated
addition (grouping) is understood.
November 2016
GROUPING ITP
If I have got 4, how many groups of 2 have
I got?
Counting forwards and recording
on a number line.
All answers to be recorded in a
number sentence following any
informal recording.
COUNTING ON AND BACK
ITP
4+8
=
Recording in number sentences and
communication along number lines
or with informal written methods.
+2
+2
______________________________
0
2
4
Know that 6 can be thought of as 5
and 1.
Counting in 2s, 5s and 10s and begin counting
in 3s and 4s.
Know that 8 is 5 and 3 therefore
subtract 5 then 3.
7
Grouping and sharing, practically. (NB If
the answer is in the same units as the
dividend, it is sharing. If the answer is in
different units, it is grouping.)
Example:
Dividend ÷ Divisor = Quotient
£20 ÷ 4
= £5 (Sharing)
£20 ÷ £4
15 – 8 =7
=
Reordering – biggest number first.
8+4
Check with the inverse.
_______________ __
10
15
leading to
= 5 (Grouping)
Introduce the x symbol once repeated addition is Record sharing by using pictorial notation
understood.
There are 6 cakes and 2 children. How
many cakes will they each get?
One for you and one for you
-3
-5
_______________________
7
10
15
Children to show notation
Find the mathematical difference
by counting on along a number line.
Share
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leading to
leading to
15 – 8
=
5 hops in 15.How big is each hop?
Reinforce the role of the number
sentence.
15 ÷ 5
=3
15 shared between 5
+2
+5
_________________________________
8
10
15
Children to show notation
Addition is the inverse of
subtraction.
Fact family.
Children to show notation
Use patterns to find answers to
subtractions
10 + 4 =
5+5+5+5+5+5
5
5
6
6
=
30
× 6 = 30
multiplied by 6
groups of 5
hops of 5
There are 7 cakes and 2 children. How
many cakes will they each get?
=
20 + 4 =
20 – 4 =
10 – 4
=5
3+2 =5
5–3=2
5–2=3
2+3
Using shapes to represent a missing
number.
1 group of 3
2 groups of 3
EXCEL PATTERNS OF
CALCULATION
Decision making
17 -
= 12
Sam works out
17 – 5 = 12.
How could he have done this?
=3
‘Left overs’ introduced.
=6
Doubles and grouping recorded on number lines
2+2
=
+2
+2
________________________________
0
2
4
Children to show notation
There are 20 sweets in a bag. How many
children can share them if they each have
5?
+5
+5
+5
+5
_____________________________________
0
5
10
15
20
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Adding more than two numbers
Strategy to include looking for
bonds that are useful eg bonds up
to and including 10, doubles or
adding 10 to a given number.
2+2+2
+2
=
+2
+2
___________________________________
0
2
4
6
-5
-5
-5
-5
_____________________________________
0
5
10
15
20
20  5
=4
Children to show notation
Introduce the x symbol once repeated addition is “How many groups of 5 are there in 20?”
understood.
Children to show notation
Compensation strategy
3x2
=
6
3 multiplied by 2 equals 6.
3 times 2 equals 6
3 lots of twos
5+9=
5 + 10 – 1
+10
____________________-1___________
5
14 15
Children to show notation
Doubles then near doubles
5+6 =
5+5+1
7+8
= 11
=
8+8 -1
= 15
In the example above with 5 rows and 9
columns, when you select to count along the
columns the given calculation is:
5 x 9 = 45 [the 5 is multiplied by 9].
Selecting to count along rows gives:
9 x 5 = 45 [the 9 is multiplied by 5].
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Decision making
Using statements such as
Ben did 14 + 9 = 23
How could he have done it?
To know that the = sign means ‘the
same as’ and can appear in a
different place within a
calculation;
E.g.
14 = 8 + 6
Addition
Emphasis on mental calculation
Subtraction
Place value, partitioning and
recombining.
Combining sets to make a total.
Progression in use of informal
recording including the number line.
Answers to be recorded as part of a
number sentence.
Reordering strategy.
COUNTING ON AND BACK ITP
NUMBER LINE ITP
Rearranging of numbers so that 36
can be seen as 30 and 6 or as 20
and 16.
Stage C (Y2/3)
Multiplication
Division
Using tables facts 2s, 10s, 5s, 3s and 4s.
Understand division as repeated addition,
grouping
Be able to partition a 2 digit number.
Table facts (see multiplication)
MULTIPLICATION BOARD ITP
Division facts corresponding to the 2, 10, 5,
3 and 4 times tables
MULTIPLICATION TABLES ITP
Partitioning of numbers into T and U
then HTU. Know what each digit
Doubles are same as x2
represents.
Vocabulary of double, multiply, groups of,
sets of, lots of etc.
TU – TU
HTU – TU
Partitioning strategy for doubling
Double 35
Use x and ÷ signs
MULTIPLICATION AND DIVISION TRIOS
SPREADSHEET
Count a handful of beads by grouping them
in fives. How many groups of 5 are there?
How many are left? Can you write a
division sentence to describe this?
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24
+
30 x 2
58
5x2
How many lengths of 6 m can you cut from
48m of rope? Write the number fact that
represents this. How did you work it out?
Record using the correct division symbol
adding in 10s and 1s
Use of number lines to record repeated
addition.
Practical apparatus to support concept.
add 20, bridge the 10
Practical contexts to be used so that the
calculation is not in the abstract.
Grouping
add 20 and then 4
Record partitioned steps in number
sentences underneath each other
and add mentally.
Use of inversion loops or the bar
model (these provide a good visual
representation of the inverse
relationship).
A lolly costs 21p. How much do 3 cost?
_________________________
0 2
4
6
8
24 + 58
20+50 =70
4+ 8= 12
24 + 58 =82
8÷2
=4
“How many groups of 2 are there in 8?”
Introduce column addition without
crossing the boundary
24 (20+4)
+53 (50+3)
77 (70 + 7)
_______________________________
0 2
4
6
8
PLACE VALUE ITP
Decision making
The number of jumps tells you the number
of groups.
Children investigate statements and solve
word problems using appropriate methods
such as mental/ jottings/ numberline.
DOUBLING AND HALVING SPREADSHEET
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Check answers by repeating addition
in different order or by an
equivalent calculation.
Compensation strategy
34 + 9
=
+10
-1
________________________
34
43 44
Partitioning the second number
strategy
76 – 43 =
76 – 40 = 36
36 – 3 = 33
73 – 46 =
73 – 40 = 33
33 – 6 = 27
Statements and word problems (mental/
jottings/ numberline)
16  2
=
“How many groups of 2 are there in 16?”
“I know that dividing by 2 is the same as
halving.”
Jump size depends on knowledge and
confidence of child. (See D)
20  5
=
Near doubles
13 + 14 =
Double 14 = 28
28 – 1 = 27
or
Double 13 = 26
26 + 1 = 27
PLACE VALUE DOTS EXCEL
SPREADSHEET
EXCEL MISSING SIGNS AND
NUMBERS
55 – 27
4 jumps
or moving away from
Counting back (left) from the larger
number in partitioned steps of the
smaller number to reach the unknown.
Partitioning the 27 into 20, 5 and 2.
←
Adding zero leaves a number
unchanged/ adding ten to a
number keeps units digit constant.
Decision making (mental, jottings,
-2
-5
-20
_____________________________________
28 30
35
55
55 – 27
+ notation
2 double jumps because 5x2
=10
Decision making
Children investigate statements and solve
word problems using appropriate methods
such as mental/ jottings/ numberline.
= 28
November 2016
numberline)
Statements and word problems.
or
or
Find the difference (counting on to
the right)
55 – 27 = 28
“How many more do I need to add to
27 to get to 55?”
+3
+20
+5
________________________________
27
30
50
55
Subtract mentally pairs of multiples
of 10 and 100, using known facts
60 – 20
=
700 – 300
40 because 6 – 2
=
=4
400
Continue to use the vertical number
line.
November 2016
Use of apparatus (Diennes)
to understand rearrangements, eg 72
as 60 and 12, not as part of
calculations.
BEADSTICKS ITP to be used with
diennes to develop concept of exchange.
PLACE VALUE ITP
Decision making
Statements and word problems.
Addition
Counting on in multiples of 100s,
10s or units using a number line.
HTU + TU, HTU + HTU.
Cross the 10s/100s boundary.
Subtraction
Stage D (Y3/4)
Multiplication
Division
Counting backwards and forwards
beyond zero, negative and positive
numbers.
Known table facts 2,3,4,5,6, 8 and 10.
Know all for 2,3,4,5,6, 8 and 10.
NUMBER DIALS ITP
Know what each digit represents in a HTU
number
-5 is negative 5 and minus 5
MULTIPLICATION TABLES ITP
November 2016
TU – TU, HTU – TU, HTU – HTU
Lead on to decomposition method in
expanded format.
NUMBER BOARDS
(all stages onwards for a range of
numbers)
Start with least significant digit
67
+ 24
11 (7+4)
+ 80 (60+20)
91
Ensure understanding of number
partitioning and exchange.
Use of apparatus (Diennes)
to understand rearrangements, eg 72
as 60 and 12, not as part of
calculations.
“7 add 4 equals 11 and 60 add 20
tens
Know what each digit represents, partition a
three digit number
Commutative law (the principle that the
order of two numbers in a multiplication
calculation makes no difference, e.g.
5x7=7x5).
MOVING DIGITS
Consolidate arrays and repeated addition.
Recalling facts.
=
5x4=
=
1 and 1 ten + 8
= 9 tens”
625
+ 48
13 (5+8)
2
12
22
32
42
52
62
Least significant digit is always dealt
with first to establish if the exchange
is needed.
Check for mental approach first
before written method. “Can I do this
in my head?”
Use numbers that will generate remainders.
‘r’ notation for the remainder.
21 ÷ 5 = 4 r 1
4x5
70
60
50
40
30
20
10
equals 80. 1+ 0
Multiply by 10 / 100, understanding the
shift in the digits
20
20.
Use partitioning/re-arranging to find
multiples of the divisor.
Partitioning method
48  3
‘What do I know about the divisor/3 x
tables?’
“I know 3 x 10
Informal recording of partitioned numbers
6x3
30.”
=
= 16
18
10 x 3
=
30
= 75
= 50
5 x 5 = 25
10 x 5
27 x 3
=
30 18
 
10 6
48  3
15 x 5
=
Decision making
= 81
20 x 3
= 60
November 2016
60 (20 + 40)
+ 600 (600 + 0)
673
NUMBER BOARDS (all stages onwards
for a range of numbers)
7x3
=
21
Reduction Strategy
All language in the context of the
place value and the mental addition
of the totals to be done in any
order.
625
+ 48
673
Decision making.
783 – 356
300, 50 and 6
-6
-50
427 433
483
783 – 356 = 427
-300
783
Difference strategy
Moving from left to right
“ 20 multiplied by 3 equals 60 and 7
multiplied by 3 equals 21. 60 add 21 equals
81.”
23 x 8 =
20 x 8 =160
3 x 8 = 24
“How many more do I need to get
from 356 to 783?”
Leading to
Both strategies need to record the
answer in a number sentence
783 -356
23
x 8
24 (8 x3)
160 (8 x20)
184
Decision making
= 427
November 2016
“783 to subtract 356 equals 427”
89 = 80
- 24 = 20
9
4
60
5
=
65
“9 subtract 4 equals 5 and 80
subtract 20 equals 60. 60 and 5
make 65”
Vertical number line
24
+6
30
+50
80
+9
89
“Add 6 to 24 to make 30. Add 50
to 30 to make 80. Add 9 to 80 to
make 89. So 6 add 50 add 9 equals
65.”
81 = 80
1
- 57
50
7
=
__
______________
“ 1 subtract 7 is tricky so I will
rearrange 81 into 70 and 11. 11
subtract 7 equals 4 and 70 subtract
50 equals 20. 20 and 4 make 24.”
81 = 70
- 57
50
___
20
11
7
4
= 24
November 2016
“I’m going to exchange a _____ for a
_____.”
BEADSTICKS ITP
Decision making
Addition
Continue with HTU + HTU, then
extend to ThHTU + ThHTU.
Approximate using the most
significant digit, rounding skills.
Check using the inverse.
Refer to the carried digit as a ten or
a hundred.
Subtraction
Stage E (Y4/5)
Multiplication
HTU – TU, then HTU – HTU
Know table facts up to 12 x 12.
(ThHTU – ThHTU)
Approximate first.
Know division facts corresponding to tables up
to 12 x 12
Approximate first using multiplication facts.
(THHTU – HTU)
Extend to simple decimals with or
without exchange from pence to
pounds.
Partitioning / distributive law, e.g. 28 x 4
can be split up into 25 x 4 add 3 x 4 or 30
x 4 subtract 2 x 4.
Moving digits ITP
Ensure that the setting out is
accurate.
754 – 86 = 668
Take away
(left)
Recap on the finding of remainders on the
number line.
Pupils to explain the effect of multiplying by
10 and 100
Addition to be done mentally.
HTU and TU x U
-6
-80
____________________________________
668
674
754
Divide any integer up to 1000 by 10
“900 ÷ 10 = 90 because the digits move one
place to the right”
Ensure that all calculation is
checked before started for any
other possible ‘tricky’ bits.
“7 add 5 equals 12. That’s 2 units
and 1 ten to carry over. 80 add 70
equals 150 and the one ten to carry
makes 160. That’s 6 tens and 100 to
carry over. 500 add 400 equals
900 and the one hundred to carry
makes 1000”
Division
Record using grid notation and expanded
short multiplication.
21 ÷ 5
“What do I know? I know that 21 is not a
multiple of 5, so there will be a remainder.”
DIVISION WITH REMAINDERS
Use problems in contexts that require the
November 2016
answer to be rounded up or down following
the remainder.
Or
Eg 35 children to sleep four to a tent. How
many tents do we need?
Or
Find the difference (right)
+4 +10
+654
__________________________________
86 90
100
754
Decomposition
(Continue with Diennes and/or
money as appropriate)
754
- 86
700
600
50
80
60
4
6
8
346
x9
54 (9 x 6)
360 (9 x 40)
2700 (9 x 300)
31 1 4
11
Continue to use partitioning/re-arranging
method.
Decision making
79
70
60
50
Children investigate statements and solve
word problems using appropriate methods.
Children are also given examples of x9 and
encouraged to think about using methods such
as x10 and subtracting x1.
79 ÷ 3
=
‘What do I know about the divisor/3 x
tables?’
“I know 3 x 10 = 30.”
20
÷3
9
19
29
=
6 r1
Fractions relate to division. ÷ 4 is the same
as halve and halve again.
= 668
Approximate answer first using multiplication
facts
Recognise that division is non-commutative.
754
- 86
600 140 14
80 6
600
60 8
Know that a number cannot be divided by 0.
=
668
Begin to use informal method for TU by U
November 2016
division (additive chunking/chunking up).
Informal recording for division
NUMBER BOARDS
“It’s tricky to take 6 from 4 and
80 from 50. I need to rearrange
the number. I will exchange one ten
from 50 which leaves 40 and
makes 14 in the units. 40 to
subtract 80 is tricky. I will exchange
one hundred from 700 and make
140. 14 subtract 6 equals 8. 140
subtract 80 equals 60 and 600
subtract 0 equals 600.”
Decomposition
£
8.95
-4.38
=
£
8 . 80 15
4 . 30 8
4 . 50 7 =4.57
96  6
“What do I know? 6 x 10 = 60”
60 36
 
10 6
96  6
= 16
“What do I know?
Set up partial multiple table:
1
2
4
10 
5
6
12
24
60
30
84 ÷ 7 =
80
+
4
70
+
14
10 + 2
= 12
November 2016
Emphasis on language of place
value, i.e. 14 units subtract 6 units,
14 tens subtract 8 tens, and 6
hundreds subtract 2 hundreds.
Use informal method for ÷ with bigger
numbers using knowledge of multiples.
387 ÷ 6
=
380 + 7
370 + 17
360 + 27
= 64 r 3
60 + 4
‘I know that 6 x 6
360’.
=
36 so 60 x 6 will be
Moving towards using known facts to
recognise multiples.
387 ÷ 6
=
360 + 27
60 + 4
= 64 r 3
Begin to use Short division with ‘bus stop’
notation
November 2016
“483 divided by 7. 4 hundreds cannot be
shared equally between 7, so exchange the
100s for 40 tens. I now have 48 tens which
shared equally between 7 is 6 with a
remainder of 6 tens. Exchange the 6 tens
for 60 units, we now have 63 units.
63 divided equally between 7 equals 9. The
answer is 69.”
Use Diennes or place value equipment to
model
Addition
Subtraction
Add with increasingly large numbers
using the compact method.
Extend methods to include decimals to
two decimal places.
Subtract with increasingly large
numbers using the compact method.
+
124.94
7.25
132.19
11
Stage F (Y5/6)
Multiplication
Th HTU , HTU , TU x TU and U
Division
Know division facts corresponding to tables up
to 12 x 12 and be able to apply them.
HTU x U
Extend methods to include decimals
to two decimal places.
TU x TU
Use the relationship between multiplication
and division
28 x 27
Dividing up to 10,000 by 10/100
Check with inverse operation. Use of
November 2016
calculator.
Continue to use: Short division with ‘bus stop’
notation
Addition to be done mentally or across
followed by column addition.
28
x 27
56 (7x8)
140 (7 x20)
160 (20x8)
400 (20x20)
756
28 X 27
= 756
£2.73 x 3
£2.73 x 3 = 273p x 3
Followed by appropriate addition calculation.
273p x 3
= 819p = £8.19
“483 divided by 7. 4 hundreds cannot be
shared equally between 7, so exchange the
100s for 40 tens. I now have 48 tens which
November 2016
4346 x 8 = 34768
32000
2400
320
+
48
34768
4346
x
8
48 (8x6)
320 (8x40)
2400 (8 x300)
32000 (8x4000)
34768
Leading to short multiplication
shared equally between 7 is 6 with a
remainder of 6 tens. Exchange the 6 tens
for 60 units, we now have 63 units.
63 divided equally between 7 equals 9. The
answer is 69.”
Use Diennes or place value equipment to
model
Express the quotient as a fraction or
decimal.
e.g.
486 ÷ 8 = 60 r 6
Fraction
60 6
8
Decimal 60.75
DIVISION WITH REMAINDERS PPT
(example given below)
17 ÷ 5
“What do I know? 17 is not a multiple of 5”.
November 2016
3 2 = 3.4
5
From knowledge of decimal/fraction
equivalents or by converting two fifths into
four tenths
Decision making
Children investigate statements and
solve word problems using appropriate
methods. Children investigate
alternative methods such as
compensation strategies and doubling
and halving and discuss when
these might be most appropriate
and efficient.
Examples:
24x99 could be done using the grid
method, but could also be calculated
by x100 and subtracting 24x1.
24 x25 could be done using the grid
method, but could also be calculated
Decision making
Word problems, e.g. 200 people attended a
concert. 1/5 of the people had complimentary
tickets. The rest paid £7.50 each. How much
money was collected from selling tickets?
Money and measures, e.g. Which is longer:
3
/4 of an hour or 2500 seconds?
Every day scenarios, e.g. Peter's family have
a meal out to celebrate his birthday. The
meal costs £52 and the restaurant adds a
15% service charge. How much is the bill
altogether
November 2016
by 24x100, halving to find x50 and
halving again to find x 25.
or using doubling and halving,
24 x25=12x50
=6 x100
Partitioning method for HTU
847 ÷ 7
“What do I know? I know 7x12 = 84
so 7 x 120
= 840”
Problem solving
Work out 575 ÷ 25, explaining your
method.
Peter says that, if you want to divide a
number by 12, you can divide it by 4 then by
3. Is he right? Explain how you know. Work
out 768 ÷ 12 using Peter's method and
using another method. Do you get the same
answer?
How many 35p packets of stickers can I buy
with £5? Explain how you know.
Coaches have 56 seats for passengers. How
many coaches are needed to take 275 people
on a trip?
November 2016
Complete this calculation: 943 ÷ 41
=2
Work out whether or not 29 is a factor of
811.
Addition
Promote decision making so that
pupils choose an appropriate
method/strategy.
Continue the use of informal
methods and number lines.
Ensure understanding of standard
written method.
Practice ThHTU + ThHTU then
calculations with any number of
digits.
Subtraction
ThHTU – ThHTU then any number
of digits.
Appropriate use of a calculator
including interpretation of displays.
Money, measures and real life
contexts.
6467 – 2684
Approximate using the most
significant digits and have a feel
for the ‘whole’ number.
Stage G (Y6)
Multiplication
ThHTU x U and HTU x TU
Division
Multiplication of decimals
Know all multiplication facts and
corresponding division facts to 12 x 12 and
beyond and be able to apply them.
TU x TU
Explain the effect of dividing by 000
78
x 42
16 (2 x 8)
140 (2 x 70)
320 (40 x 8)
+2800 (40 x 70)
3276
1
Extend methods to include HTU ÷ TU
Continue to use the short division method
when the two digit divisor is up to 12 or is a
easily recognisable multiple eg 20, 25 or 50.
Compact (long)
http://mathsonline.org/pages/longmult.html
Appropriate teaching/use of the
calculator including interpreting the
display, eg money or measures.
Calculator display 0.37 is then
interpreted as 37p in the context of
money. Remind 4p = 0.04
Calculator display £1.4 is interpreted
as £1.40
324.9 – 7.25
78
X42
156
1
+ 3120
3276
Use a calculator appropriately, approximating
first
Use of calculator for interpreting the
November 2016
7648
+ 1486
9134
111
Decimal points are fixed on the line
with digits in the squares.
124.9 + 7.25
124.90
+
7.25
132.15
11
quotient by entering a fraction to find the
decimal equivalent.
=
13 .6 – 2.8
http://mathsonline.org/pages/longdiv.html
+0.2
+10.6
__________________________________
2.8
3
13.6
Use long division only with pupils who are
secure with number sense and place value.
384  16
13.6 – 2.8
= 10.8
“What do I know about the divisor?”
Record partial tables.
leading to
Decimals - fill ‘empty columns’ with
zeros.
Long division
14.6 – 2.76
__24
16 ) 3864
– 32 
64
(thinking not generally
recorded)
(38÷16=2 r 6;
2x16=32)
(bring the 4 down)
(16 into 64=4;
4x16=64)
– 64
0
(no remainder)
Involve decimals, money and measures
through approximation and appropriate use
of the calculator.
Addition either mentally or by column
addition
Decimals. Teach children how to use known
November 2016
facts to build new facts according to the
place value required, e.g.
3x4
=12
3x0.4 = 1.2
3x0.04 = 0.12
0.75 x 6
= 4.2
0.05 x 6 = 0.3
0.75 x 6 = 4.5
0.7 x 6
Grid method based upon very secure place
value.
Problem Solving
Mike works out that 14 × 12 = 168. What is
14 × 1.2? How do you know?
Use a written method to calculate 24 × 13.
What do you need to change to show a
similar method to work out 2.4 × 13?
Use a written method to find the area of a
swimming pool which is 25 m long and 7.5
m wide.
Complete the missing sections to work out 35
Continue to make use of partitioning and the
number line for repeated addition where
appropriate.
944 ÷ 22 =
November 2016
× 2.1:
Which is closer to 100: 5.2 × 17 or 7.2 ×
15? Use written methods to prove your
answer.
What do I know about the divisor?
440 ( 22 x 20)
880 (22 x 40)
944 ÷ 22 = 42 r 20
Express the remainder as a fraction or
decimal.
Problem Solving :
Division giving a decimal answer, e.g. Divide
9 by 5 giving your answer as a decimal.
Missing number calculations, e.g.
= 0.04; 0.6 ×
÷8
= 4.2
November 2016