Calculation Policy - Southwood Infant School

Written by Becky Blamires

Calculation Policy


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


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


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.


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?


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.


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.


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


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


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


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.


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


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


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


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 …


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


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.


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


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.


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


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


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


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?


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.


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


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.


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.


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


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



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


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.


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


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


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:


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.

Calculation Policy - Southwood Infant School

Documents

Transcript of Calculation Policy - Southwood Infant School