Welcome to Southwood! Open House 2012-2013 Mrs. Mar, Room 244.
Calculation Policy - Southwood Infant School
Transcript of Calculation Policy - Southwood Infant School
Written by Becky Blamires
Calculation Policy
Written by Becky Blamires
Maths at Southwood
At Southwood Infants we aim to instill a love of Maths and encourage children to become
confident learners.
We plan our maths in accordance with the requirements of the new national curriculum. Most
of this is taught in a daily numeracy lesson, but additional opportunities for using and applying
and developing investigational strategies are also provided.
It is important that the children develop a confident and positive attitude towards
Mathematics and understand its use in everyday life. Mathematics equips children with a
powerful set of tools to understand and change the world, such as logical reasoning and
problem solving skills. Problem solving should be about valuing independence and individual
ideas, and being given opportunities to be creative and develop creative thinking through –
Appropriate practical activities
The process of enquiry and investigation and a systematic approach
Mathematics skills and knowledge and a quick recall of basic facts
An ability to identify patterns, relationships and generalise in mathematics
An awareness of the uses and applications of mathematics in everyday language
The ability to express ideas concisely using accurate mathematical language
The ability to select and use a range of mathematical tools
An enjoyment of mathematics
At Southwood we aim to build essential mathematical knowledge, develop and embed key skills
and challenge and extend the children’s thinking. Maths is taught throughout the school using a
range of strategies including whole class, group and individual teaching. Children are given the
opportunity to do practical investigative and written work. There is an emphasis on mental and
oral work using a wide range of strategies. We make learning experiences enjoyable,
motivating and exciting and encourage all children to participate and celebrate perseverance
and resilience as important skills. We provide open ‘rich tasks,’ open-ended problems for the
children to solve, which help to foster an enquiring mind where Maths is seen as a challenge
rather than a chore.
The only way to learn mathematics is to do mathematics.
Paul Halmos
Written by Becky Blamires
Overview
The ability to calculate mentally lies at the heart of the mathematics
taught at primary school. During Key Stage 1, emphasis will be placed
upon developing mental calculations. Written recordings are used to
support and develop these mental strategies.
Children will always be encouraged to look at a problem and then decide
which method is the best to use. They should ask themselves…
‘Can I do this in my head?’
‘Can I do this using drawings or jottings?’
In Year R, children’s understanding of mathematics develops through
rich, child initiated play and first hand experiences.
We provide a mathematically rich environment, both indoors and outdoors
where children are able to access resources independently all of the time.
This is achieved by:
· Direct Teaching of number recognition, accurate counting skills,
sharing, sequencing, shape recognition and pattern making through
practical activities.
· Enhancing their play with topic based mathematics and role play.
· Maximizing the mathematical potential of classroom routines. For
example,
How many children are having red/blue school dinners?
How many altogether?
What is the day today? What was yesterday/tomorrow?
What is the date? Yesterday/tomorrow? How do you know?
In Year 1 and Year 2 the children’s understanding of the properties of
numbers is developed, this vital understanding gives the children a range
of strategies, from which to choose the most effective to solve all kinds
of number problems, for example:
Learning all the pairs of numbers that make 10: We call these
number bonds.
0+10, 1+9, 2+8, 3+7, 4+6, 5+5.
Learning all addition doubles to 10: 1+1, 2+2 etc to 10+10 and then
the corresponding halves: half of 10, half of 8, etc
Written by Becky Blamires
Learning what each digit is in a two digit (and then three digit)
number represents: 24 is two tens and four ones, etc
Learning to add and subtract 1 and then add and subtract 10.
Children will begin to learn their times tables, counting in 2’s, 5’s and
10’s
When teaching these key skills, whenever possible, they are taught in
context, for example using money or units of measure and presented to
the children as a problem to solve. The children are encouraged to think
of ways to solve a problem and to notice any patterns as they are
working.
Children first work practically and are encouraged to use a large range of
apparatus (numicon, bead strings, counters, cubes, dienes etc) then they
record using pictures and finally record using more formal number
sentences.
As the children become more confident they are expected to explain to
others how they have solved a problem.
Teachers will challenge and extend the children’s understanding by
asking them further challenging questions as they work.
Written by Becky Blamires
Glossary of mathematical terms
Bridging: when children cross a boundary e.g. multiples of 10,
100, 1000. We refer to this as bridging e.g. adding 8 onto 17
children will add 3 to 20 then add 5 (the children have
partitioned
8 into 3 and 5 and used 20 as a bridge).
+3 +5
17 20 25
Number line (structured): a line marked with numbers.
Number line (unstructured): a blank line that numbers can be
written on.
Number sentence: mathematical sentence written in numerals
and mathematical symbols e.g.
3 x 7 = 21, 6 + 3 = 9, 10 – 2 = 8, 15 ÷ 3 = 5
Numicon – a practical multi sensory resource
Partition: to partition a number means, ‘breaking the number up
in different ways.’ The most common way to partition in primary
school is into hundreds, tens and ones e.g. 472 = 400 + 70 + 2
but numbers can also be partitioned in different ways e.g.
8 = 7 + 1, 6 + 2, 5 + 3, 4 + 4.
Place value: the value of a digit depending on its place in a
number e.g. 354 the value of the 4 is four ones*, whereas in the
number 435 the 4 has a value of four hundreds.
*Please note that we no longer use the term ‘units’ but say ‘ones’
instead.
Single digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are single digit
numbers.
Word problem: a calculation put into a context e.g. 5 + 10 as a
word problem could be, ‘Mary has a 5 pence coin and a 10 pence coin. How much does she have altogether?
Written by Becky Blamires
End of Year Expectations for addition and subtraction
Year 1 Year 2
Pupils should be taught to:
read, write and interpret
mathematical statements
involving addition (+),
subtraction (-) and equals (=)
signs
represent and use number
bonds and related subtraction
facts within 20
add and subtract one-digit and
two-digit numbers to 20,
including zero
solve one-step problems that
involve addition and
subtraction, using concrete
objects and pictorial
representations, and missing
- 9.
Pupils should be taught to:
solve problems with addition
and subtraction:
using concrete objects and
pictorial representations,
including those involving
numbers, quantities and
measures
applying their increasing
knowledge of mental and
written methods
recall and use addition and
subtraction facts to 20
fluently, and derive and use
related facts up to 100
add and subtract numbers
using concrete objects,
pictorial representations, and
mentally, including:
a two-digit number and ones
a two-digit number and tens
two two-digit numbers
adding three one-digit
numbers
show that addition of two
numbers can be done in any
order (commutative) and
subtraction of one number
from another cannot
recognise and use the inverse
relationship between addition
and subtraction and use this
to check calculations and
missing number problems.
Written by Becky Blamires
Addition Stage 1. Counting objects.
Children are encouraged to develop a mental picture of the
number system in their heads to use for calculation. They
develop ways of recording their own calculations using pictures
and images to develop their mathematical thinking. Ensure the
mathematical language of ‘addition is used
Ie
Match objects to numicon pieces
Always in the context of a problem/story:
I find 3 shells and my friend finds 4 shells. How many shells did we find
altogether?
then
Begin by counting the objects first. Count out the objects:
1, 2, 3, 4, 5… 1, 2, 3, 4
Then put them all together and count the objects again,
from the start: 1, 2, 3, 4, 5, 6, 7, 8, 9
The next step will be to start at the first number they have and
then count on (so for 5 add 4 more, children could start from the
smaller number): e.g. 5, 6, 7, 8, 9
The next step would be to recognise that 5 is the larger
number and to count on 3 more from there: 5... 6,7,8
Add small objects in the numicon that can be counted out
or
Written by Becky Blamires
Addition Stage 2. Using a number track/ number line.
Using a number track/floor tiles, children will put objects
onto it, counting objects on each square.
They will also count using other objects, like
a bead string.
Below six beads have been counted and
marked with a peg.
Next, they will move to a structured number line, placing
objects on the line to show what we count on: e.g.
6 elephants + 3 elephants, find 6 on the number line, then
place the objects to show what we are adding.
They can begin to use the addition sign + to record
They will use numicon pieces to make the number sentence and
add together
or
Written by Becky Blamires
Addition Stage 3. Using structured and unstructured number lines.
Making pairs of numbers that make ten – mentally,
practically and written
Start with adding/ jumping 1 more, 2 more to become confident and
accurate using a structured numberline use numicon or small apparatus
to show adding on.
Using a numberline to jump in efficient jumps use apparatus to support
18+5 =
18 + 5 = 23
Explore adding two numbers on a structured numberline. They
will start to realise that it is more efficient to a start with
the biggest number on a structured number line, then count
the jumps:
7 + 11
7 + 11 = 18
Written by Becky Blamires
Partially numbered line
Using the number line to support other concepts Support counting forwards in 2s, 5s and 10s.
twos
fives
tens
• Ensures smooth progression from a fully numbered line (in
ones) to the unnumbered line below.
• Supports the development of understanding of scale and
what each unnumbered marker is worth.
Unnumbered lines with markers
• Lines with markers that can indicate equal steps of any size
(eg 2, 5, 10, 100, 1000 etc).
• Don’t have to start at 0.
• Encourages children to build on what they know about
numbers and the number system in that they need to think
for themselves where the numbers might belong.
• The counting stick is an excellent and versatile example of
the unnumbered line; each end can be anything you want it to
be.
Written by Becky Blamires
Eventually, children will draw their own empty number
lines to do this.
8 + 6
8 +6 = 14
The next step comes when adding teen numbers. Initially,
children will count on in ones.
15 + 12
KEY STRATEGY-
Bridging through Ten – practically, mentally and written
Support understanding of number facts, ie: pairs of numbers that make 10.
Lots of practical resources making to ten and over
ie 8 + 5 =
or
They will be able to make ten (or a whole ten) by
partitioning a number 8 + 2 +3 = 13
Written by Becky Blamires
Empty number line
• When introducing the ENL use diennes to model and the ENL
to record
Children need to partition the number they are adding.
Start adding teen numbers
Use diennes to add 10 and record on numberline +10 then +2
When secure using diennes and ENL can just focus on using
ENL or move to place value counters
Move onto adding any 2 digit numbers
Counting on to find the difference
Support the concept of the difference between two numbers.
The difference between 21 and 26 = 5
Written by Becky Blamires
Addition
Stage 4. The Number Line partitioning - use concrete
apparatus
Start with the biggest number. Partition the smaller
number into tens and units and add it on. Sometimes, you
might partition the tens number into a more manageable
number.
e.g. 45 + 28 Partition 28 into 10 + 10 + 8
45 + 28 = 73
Or
e.g. 45 + 28 Partition into 20 + 8
Or even partitioning the units to help as well
e.g. 45 + 28 Partition 28 into 10 + 10 + 5 + 3
45 + 28 = 73
Written by Becky Blamires
Adding 3 digit numbers using ENL as before start
adding hundreds, then tens then ones.
Explore patterns in calculation, ie: pairs of multiples of 10 with totals up to 100.
Explore patterns in calculation, crossing the 100s
boundary.
There are 76 marbles in one jar and 57 marbles in another jar. How many marbles are there
altogether?
Therefore …
Then …
Then …
Written by Becky Blamires
Addition
Stage 5. Partitioning both numbers.
Split both numbers into tens and ones (and hundreds too)
56 + 38
Partition the numbers 50 + 6 30 + 8
Add the tens 50 + 30 = 80
Add the ones 6 + 8 = 14
Now add the totals together 80 + 14 = 94
For a three digit number:
259 + 174
Partition the numbers 200 + 50 + 9 100 + 70 + 4
Add the hundreds 200 + 100 = 300
Add the tens 50 + 70 = 120
Add the ones 9 + 4 = 13
Now add the hundreds and tens 300 + 120 = 420
Now add your answer to the ones 420 + 13 = 433
Written by Becky Blamires
Subtraction Children are encouraged to develop a mental picture of the number
system in their heads to use for calculation. They develop ways of
recording their own calculations using pictures and
images to develop their mathematical thinking.
Children are encouraged to work practically to
understand the concept of subtraction as taking
away, and by comparing two objects to find difference, how many more
or less e.g. Ensure the mathematical language of ‘subtraction’ is
used subtract, take away, minus, leaves, less, smaller than.
Stage 1. Taking Away using Objects. Subtraction problems should be taught in context.
Comparison of amounts more or less.
Here are six toy cars.
How many more cars are needed to make a set of
eight cars?
The Queen of Hearts made 9 jam tarts.
She gave 3 to Alice.
How many did she have left?
Count out the jam tarts, 1 to 9. Physically
take away 3 objects and count how many
are left: 1, 2, 3, 4, 5, 6.
What is the difference between the
number of grey rabbits and the number of
white rabbits?
Teacher demonstrates use of number tracks to check results
of practical activities.
Written by Becky Blamires
Subtraction Using numicon children can explore subtraction through the
context of stories and number problems
Using the numcion tiles they can count animals/small
equipment into the holes and then ‘take away’
When secure they can overlay the amount to be taken away
and count how much if left.
Explore other subtraction questions
10 take away 3 leaves 7
Also other questions 10 take
away 7 leaves 3
Written by Becky Blamires
Stage 2. Using Objects on a Structured Number Line.
Still using objects, place them on a number line, then take
the objects away (from the right hand side of the number line
only – we want to build the idea of counting back) – to find
our answer.
So, using the same problem where the Queen of Hearts has
given 3 of her 9 tarts to Alice:
Count out the objects on the number line.
Take away the 3 tarts and count how many we have left.
Begin to count back from 9 as each jam tart is taken.
Begin to model using the subtraction sign – means take
away
9 – 3 = 6
Children can decide how to record their questions.
Written by Becky Blamires
Subtraction Stage 3. Counting Back on a Number Line.
Next, they will move to using a structured number line,
without necessarily placing objects on it, but drawing the
jumps counting back in ones.
e.g. The Queen of Hearts has 16 jam tarts, but the King
takes 7. How many are left?
16 – 7 = 9 jam tarts left
Partially numbered line
Using the number line to support other concepts. They can
write their own starting number and count back showing the
jumps.
Support counting backwards in 2s, 5s and 10s.
twos
fives
tens
• Ensures smooth progression from a fully numbered line (in
ones) to the unnumbered line below.
• Supports the development of understanding of scale and
what each unnumbered marker is worth
Written by Becky Blamires
Unnumbered lines with markers
• Lines with markers that can indicate equal steps of any size
(eg 2, 5, 10, 100, 1000 etc).
• Remember to start on the right hand side with the largest
number.
• Encourages children to build on what they know about
numbers and the number system in that they need to think
for themselves where the numbers might belong.
• The counting stick is an excellent and versatile example of
the unnumbered line; each end can be anything you want it to
be.
As the children progress they will be able to draw
numberlines themselves for the same sort of calculations:
16 – 7 = 9
Written by Becky Blamires
Stage 4. Counting back on a number line in tens and ones.
If Peter Rabbit has 34 carrots, but he
eats 18 for
lunch, how many will he have left?
Children may well still be drawing a number line and
counting back in 18 steps of 1, but by then we would be
encouraging children to partition the 18, which would be 10
and 8, and to count back 1 ten and 8 ones.
Become more efficient in subtraction by counting back
Written by Becky Blamires
Stage 5. Counting On using a number line:
Only when children are ready can they begin counting on to
solve subtraction problems. (They must be confident at
adding tens from any number and counting on in ones).
We start the children off by giving them calculations which
are easier to solve by counting on rather than by counting
back.
e.g. If Lucy has £35 in her account and Daniel has £28, how
much more has Lucy saved?
Written by Becky Blamires
Stage 6 .Finding the difference
Finding the difference
e.g. Work out the difference between 46 and 18.
Children should be encouraged to solve these types of
calculation by representing both numbers initially on separate
number lines and reinforcing the language of how many more
or less, e.g.
Through modelling and discussion, explore how this can
represented as 46-18
and that complementary addition (counting on) can be a
useful checking strategy.
Children should be encouraged to decide which strategy to
use depending on the numbers involved.
Written by Becky Blamires
Stage 7 Inverse relationship
They construct sequences of calculations involving subtraction
such as: 5 – 1 = 4, 6 – 2 = 4, 7 – 3 = 4, …
They continue sequences such as:
12 – 0 = 12, 12 – 1 = 11, 12 – 2 = 10, … to build up
patterns of calculations that highlight the underlying process
of subtraction.
They begin to recognise that subtraction and addition ‘undo
each other’. Using equipment and this model they can explore
different ‘questions’ they can find using the same numbers.
e.g. 16 + 4 = 20 and 20 – 4 = 16 20 – 14 = 6
Written by Becky Blamires
Children apply their knowledge to problems; for example, they
work out how many biscuits are left on a plate of 13 biscuits
if 4 are eaten.
Using +/- and = signs
Children record addition and subtraction number sentences
using the operation signs + and –. They generate
equivalent statements using the equals sign, for example:
7 = 6 + 1
7 = 5 + 2 …etc
7 = 8 – 1
7 = 9 – 2 …etc
They recall the number that is 1 or 10 more or less than a
given number and use this to support their calculations, for
example to give answers to 12 + 1, 13 – 1 and 30 + 10 and 60 –
10.
Written by Becky Blamires
End of year expectations for multiplication and
division
Year 1 Year 2 Pupils should be taught to:
solve one-step problems
involving multiplication and
division, by calculating the
answer using concrete
objects, pictorial
representations and
arrays with the support of
the teacher.
Pupils should be taught to:
recall and use
multiplication and division
facts for the 2, 5 and 10
multiplication tables,
including recognising odd
and even numbers
calculate mathematical
statements for
multiplication and division
within the multiplication
tables and write them
using the multiplication
(×), division (÷) and equals
(=) signs
show that multiplication of
two numbers can be done
in any order (commutative)
and division of one number
by another cannot
solve problems involving
multiplication and division,
using materials, arrays,
repeated addition, mental
methods, and
multiplication and division
Written by Becky Blamires
facts, including problems
in contexts.
Multiplication
Stage 1. Counting objects… if there are two apples on
each plate, how many have we got altogether?
We use real objects we have in the classroom that the
children may handle. Children can also draw their
representations of the objects. They can then write the
total number next to their drawings.
In terms of language, talk about ‘groups of’ apples.
Related objectives: Count repeated groups of the same size;
Share objects into equal groups and count how many in each
group, e.g.
Add trays with small compartments for sorting to
the making area. Add collections of things: bottle
tops, sequins, threads, tiny pieces of fabric, etc.
Model sharing out the objects equally. For
example: do you all want sequins? I'll put 5
each on your trays. Can you give everybody
the same number of these? Have you got the
Written by Becky Blamires
same?
Hang up 3 bags outside for making collections. Put a number 2
on each bag. Encourage the children to collect 2 of any
treasured object in each bag, for example fir cones or smooth
pebbles. The collections could be used inside and outside in
the learning environment for different purposes, for example
as a gallery of natural objects or for adding to the making
area.
Encourage children to record their own thinking.
Written by Becky Blamires
Stage 2. Repeated addition Counting in groups of…
If the elephants came in two by two, if there were 5 groups
of 2, how many elephants were there? Use practical
equipment to model the problem
Still using the objects, we can place them on a number
line and count the pairs, beginning to link to tables facts:
Children start to count up in 2s from this stage.
Children can choose how to record these questions.
Then can start to record this as repeated addition
2 + 2 + 2 + 2 + 2 = 10
Written by Becky Blamires
Use pictures and symbols to record.
There are three sweets in one bag.
How many sweets are there in five bags?
Using numicon to show repeated addition and groups of the
same
Number tracks / Number line
(modelled using bead strings)
2 x 3 or 3 x 2
[two, three times] or
[three groups of two]
Children will experience equal groups of objects and will
count in 2s, 5s and 10s and begin to count in 5s.
They will work on practical problem solving activities
involving equal sets or groups, e.g. Count five hops of 2
Written by Becky Blamires
along this number track. What number will you reach?
There are 4 apples in each box.
How many apples in 6 boxes?
Stage 3. Arrays
Arrays are useful models for multiplication which can be used in a
variety of ways, ranging from highly structured lessons to games and
open investigations.
An array is formed by arranging a set of objects into rows and
columns. Each column must contain the same number of objects as the
other columns, and each row must have the same number as the other
rows.
The following array, consisting of four columns and three rows, could
be used to represent the number sentence 3 x 4 = 12, 4 x 3 =12, 3 + 3 +
3 + 3 = 12 and 4 + 4 + 4 =12.
Children arrange items in groups:
3 rows of 5
conkers
5 + 5 + 5
5 x 3 = 15 conkers
And explore other ways of arranging them:
Written by Becky Blamires
Mastery and mastery with greater depth
Integral to mastery of the curriculum is the development of deep rather than superficial
conceptual understanding. ‘The research for the review of the National Curriculum showed that it should focus on “fewer things in greater depth”, in secure learning which persists, rather than relentless, over-rapid progression.’ 6 It
is inevitable that some pupils will grasp concepts more rapidly than others and will need to be
stimulated and challenged to ensure continued progression.
However, research indicates that these pupils benefit more from enrichment and deepening of
content, rather than acceleration into new content. Acceleration is likely to promote
superficial understanding, rather
than the true depth and rigour of knowledge that is a foundation for higher mathematics.
Within the materials the terms mastery and mastery with greater depth are used to
acknowledge that all pupils require depth in their learning, but some pupils
will go deeper still in their learning and understanding.
Mastery of the curriculum requires that all pupils:
use mathematical concepts, facts and procedures appropriately, flexibly and fluently;
recall key number facts with speed and accuracy and use them to calculate and work
out unknown facts;
have sufficient depth of knowledge and understanding to reason and explain
mathematical concepts and procedures and use them to solve a
A useful checklist for what to look out for when assessing a pupil’s understanding might be:
A pupil really understands a mathematical concept, idea or technique if he or she can:
describe it in his or her own words;
represent it in a variety of ways (e.g. using concrete materials, pictures and
symbols – the CPA approach)
explain it to someone else;
make up his or her own examples (and non examples) of it;
see connections between it and other facts or ideas;
recognise it in new situations and contexts;
make use of it in various ways, including in new situations.
Developing mastery with greater depth is characterised by pupils’ ability to:
solve problems of greater complexity (i.e. where the approach is not
immediately obvious),
demonstrating creativity and imagination;
independently explore and investigate mathematical contexts and structures,
communicate results
clearly and systematically explain and generalise the mathematics.