greenfields federation littlehaven infant school northolmes junior school

Greenfields FederationLittlehaven Infant SchoolNortholmes Junior School

Aims

The national curriculum for mathematics aims to ensure that all pupils: •become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. •reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language •can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more formal written methods of calculation.

Strategies for calculation need to be represented by models and images to support, develop and secure understanding. This, in turn, builds fluency. When teaching a new strategy it is important to start with numbers that the child can easily manipulate so that they can understand the methodology.

The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. Previous stages may need to be revisited to consolidate understanding when introducing a new strategy.

A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.

Introduction

Magnitude of CalculationsReception– Children count reliably with numbers from one to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing. Year 1 – U + U, U + TU (numbers up to 20) including adding zero, U – U, TU – U (numbers up to 20) including subtracting zero, U x U, U ÷ U

Year 2 - TU + U, TU + multiples of 10, TU + TU, U + U + U, TU - U, TU – tens, TU – TU, TU x U, U ÷ U

Year 3 – add numbers with up to three-digits, HTU + multiples of 10, HTU + multiples of 100, subtract numbers up to three-digits, HTU – U, HTU – multiples of 10, HTU – multiples of 100, HTU – HTU, TU x U, TU ÷ U

Year 4 - add and subtract numbers with up to four-digits, ThHTU + ThHTU, ThHTU - ThHTU, add and subtract decimals with up to two decimal places in the context of money, multiply three numbers together, TU x U, HTU x U, TU x U, multiply by zero and one, TU ÷ U, HTU ÷ U

Year 5 – add and subtract numbers with more than four-digits, add and subtract decimals with up to three decimal places, ThHTU x U, ThHTU x TU, HTU x TU, multiply whole numbers and decimals with up to three-decimal places by 10, 100 and 1000, divide numbers with up to four-digits by U (including remainders as fractions and decimals and rounding according to the context)

Year 6 - add and subtract numbers with more than four-digits, add and subtract decimals with up to three decimal places, multiply numbers with up to four-digits by TU, multiply numbers with up to two-decimal places by a whole number, divide numbers up to four-digits by TU (interpreting remainder according to the context), divide decimals up to two-decimal places by U or TU

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of

mathematical ideas. … pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning

and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to

science and other subjects.

National Curriculum 2014

Symbols

Concrete Experiences

PicturesLanguage

12 + 19

Active/concrete

Building visual images

Abstract13 - 8

Haylock and Cockburn (2008)

Structuring Learning

Children must have concrete experiences that enable them to

create visual images. They should be encouraged to

articulate their learning and to become pattern spotters.

Numicon

bead string

1 0

1

count stick

Cuisenaire

place value apparatus

number linenumber

grids100 and 200

Multilink

0.1 place value counters

10.110

100

double sided counters

http://www.amazon.co.uk/Learning-Resources-Mathlink-Maths-Cubes/dp/B000URL296/ref=sr_1_1?ie=UTF8&qid=1343322814&sr=8-1

AggregationUnion of two sets

How many/much altogether?The total

Structures of Addition (Haylock and Cockburn 2008)Children should experience problems with all the different addition structures in a

range of practical and relevant contexts e.g. money and measurement

AugmentationStart at and count on

Increase by Go up by

Commutative lawUnderstand addition can be done in any orderStart with bigger number when counting on

(Explain to children that subtraction does not have this property)

6 7 8 9 10 11 12 13 14 15

+1 +1 +1 +1 +1 +1 +1 +1 +1

is the same as/equal to (=)

NC End of Year Expectations Children’s Recording

Addition

Children may record pictorially progressing to

recording number sentences alongside

If using Numicon, children could use printed Numicon icons and stick these in - progressing

to recording number sentences alongside

Example

+ =

1 + 2 = 3

Year R

Fluency

Count forward in ones,

Be able to add one more

Read digits up to 20

Match written numbers to number of objects

Order concurrent numbers upto 20

Recognise and use the + symbol

Order non-concurrent numbers eg: 1, 3, 5, 9

Children count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number.

Using quantities and objects, they add and subtract twosingle-digit numbers and count on or back to find the answer.

They solve problems, including doubling, halving and sharing.

Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together.

Understand what a number looks like.. Eg what is 6? 6 bears. 6 [pencils. 6 children etc.

Use practical resources such as bears, counters, cubes and number lines/hundred grids and progress to a resource such as Numicon to

encourage counting in ones and then groups.

‘’one more than three is

four. One less than

four is three’

1 2 3 4 5 876 90

Possible Concrete and Visual Representations

10

5 + 4

http://www.google.co.uk/url?url=http://illuminations.nctm.org/Lesson.aspx%3Fid%3D355&rct=j&frm=1&q=&esrc=s&sa=U&ei=DAWtVP2RKZGtadXTgPgL&ved=0CC4Q9QEwDA&sig2=7-M7-7ulI9Zl8sLC4QeSaw&usg=AFQjCNEsUw8WTeYHu5XHsm-kocc_7d4pdA


Addition

Fluency



10

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 number problems such as 8 + __ = 13

Children must experience combining two, and then more than two, groups of objects using counting on and the language of addition e.g. add, plus

Year 1

Children must experience increasing numbers e.g. what is two more than seven ?

Compare quantities to say how many less and/or how many more

Represent and use number bonds to 5

Represent and use number bonds to ten

Add using doubles

Add numbers mentally by counting on TU + U or

U + U + U (not crossing 10 barrier)

Count forwards, to and across 100, beginning with 0 or 1 or from any

given number

Switch count between tens and ones

e.g. 10, 20, 30, 31, 32, 33 …

Represent and use number bonds up to 20 (establish addition and subtraction as related

operations)

Count, read and write numerals to 100

Read and write numbers to 20 in numerals or

words

Find one more than a number

Find ten more than a number

Count in multiples of 2s, 5s and 10s starting on multiples to highlight pattern recognition





+ =

1 + 2 = 3

9 + 6

6 7 8 9 10 11 12 13 14 15 16


encourage counting in groups rather than ones

‘two more than three is five or two

less than five is three’

+2

1 2 3 4 5 876 90

4 + 2 two more than four

2 4 6 8 10 12 14 16 18


Addition

Fluency

Year 2 Children apply, develop and secure their understanding of place value


Children should be able to partition numbers in different ways e.g. as 2+2+2+1 or 5+3 or 23 as 20 +3 or

10+13

Use jottings and record number sentences

41

+

28

40

+ 20

= 60

1

+ 8

= 9 60 + 9 = 69


10

Use Numicon, number grids, place value apparatus/Dienes, place value grids, place value cards, Encourage children to

partition numbers rather than counting in ones.

20 2

0 5 10 15 20 25 30 35 40 45

Numbered and partially numbered number lines

7 3?

?

7

Bar Model

Cuisenaire

?

Children use blank number lines for TU + TU or HTU + TU

Represent and use numberbonds to 20

Add numbers mentally by counting on TU + U or U + U + U or TU + TU(crossing 10’s barrier)

Count in steps of 2, 3, and 5 from 0, and in tens from any number, forward and backward

Recognise the place value of each digit in a two-digit number (tens, ones)

Identify, represent and estimate numbers using different representations, including the number line

Compare and order numbers from 0 up to 100; use <, > and = signs

Read and write numbers to at least 100 in numerals and in words

Use place value and number facts to solve problems.

Use knowledge to begin to derive and use number facts up to 100 (multiples of 10)

Solve problems with addition and subtraction: - using concrete objects and pictorial representations, including those involving numbers, quantities , measures, money and real life contexts -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 solve missing number problems.

Children should understand the language of sum

ENSURE CHILDREN HAVE THE OPPORTUNITY TO ADD MORE THAN TWO NUMBERS

NC End of Year Expectations Teacher Modelling/Children’s Recording

AdditionFluency

Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. They should be taught together.


10Year 3

Column addition (no exchanging) with up to three-digits

100 + 40 + 1 + 100 + 20 + 8

200 + 60 + 9 = 2 6 9

Expanded recording

4 0 + 1

+ 2 0 + 8

6 0 + 9 = 6 9

4 0 + 3

2 0 + 8

7 0 + 1 = 7 1

Children apply, develop and secure their understanding of place value and begin to record in columns

Concrete/Visual representatives SHOULD be used alongside algortihms

Expanded recording without exchange

Expanded recording with exchange1

?770

110100

100 10 3

Partially numbered and blank number lines

Cuisenaire

30Bar Model

1

Represent and use numberbonds to 100

Count from 0 in multiples of 4, 8, 50 and 100;

Find 10 or 100 more or less than a given number

Mentally add HTU + ones, HTU + tens, HTU + hundreds

Count in ones, tens and hundreds maintaining fluency through varied and frequent practice Recognise the place value of each digit in a three-digit number (hundreds, tens, ones)

Compare and order numbers up to 1000

Identify, represent and estimate numbers using different representations

Read and write numbers up to 1000 in numerals and in words

Solve number problems and practical problems involving these ideas.

Add and subtract numbers mentally, including: a three-digit number and ones a three-digit number and tens a three-digit number and hundreds

Add and subtract numbers with up to three digits, leading to using formal written methods of columnar addition and subtraction

Estimate the answer to a calculation and use inverse operations to check answers

Solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.

Children should partition numbers, up to 1000, in different ways

e.g. 100 + 40 + 6 or 100 + 30 + 16

ENSURE CHILDREN HAVE THE OPPORTUNITY TO ADD MORE THAN TWO NUMBERS WITH DIFFERING NUMBERS OF DIGITS

1 0

Children apply, develop and secure their understanding of place value and use partitioning to add horizontally

Concrete/visual representatives SHOULD be used alongside algortihms

E.g. 41 + 28 43 + 28

40 + 20 + 1 + 8 = 69 40 + 20 + 3 + 8

Or Or

41 + 20 + 8 = 69 43 + 20 + 8

End of Year Expectations

Teacher Modelling/Children’s Recording

Addition

Fluency

Year 4

Column addition (no exchanging) with up to three-digits

100 + 40 + 1 + 100 + 20 + 8

200 + 60 + 9 = 2 6 9

Column addition (with exchanging)

1 4 1 + 1 2 8

2 6 9

Expanded recordingCompact (column) recording

4 0 + 1

+ 2 0 + 8

6 0 + 9 = 6 9

4 0 + 3

2 0 + 8

7 0 + 1 = 7 1


Concrete/visual representatives SHOULD be used alongside algortihms

Expanded recording without exchange

Expanded recording with exchange

Compact (column) recording

£ 7. 8 9 + £ 6. 4 2

£ 1 4. 3 1

7 8 9 + 6 4 21 4 3 1 1 1

H T U

H T U

11


110100 1

1

100 10 3

?7

Bar Model

7030

0 10 20 30 40 50


Add decimals in the context of money

1 0

1 4 3 + 1 2 8

2 7 1

1

Cuisenaire

?

10100

1

1

200 20 3

40

Count in multiples of 6, 7, 9, 25 and 1000

Find 1000 more or less than a given number

Count forwards through zero starting with negative numbers

Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones)

Order and compare numbers beyond 1000


Add numbers mentally by partitioning (TU + TU)

Use adjusting to add mentally (45 + 19)

Perform mental calculations with two-digit numbers, the answer could exceed 100 and increasingly large numbers

Round any number to the nearest 10, 100 or 1000

Solve number and practical problems that involve all of the above and with increasingly large positive numbers

Read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value.

Add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate

Add together numbers with up to two decimal places in the context of money

Estimate and use inverse operations to check answers to a calculation

Solve addition and subtraction two-step problems in contexts, (including missing number problems)deciding which operations and methods to use and why.

ENSURE CHILDREN HAVE THE OPPORTUNITY TO ADD MORE THAN TWO NUMBERS INCLUDING DECIMALS, WITH DIFFERING NUMBERS OF DIGITS

End of Year Expectations Teacher Modelling/Children’s Recording

Addition

Year 5

Fluency

Column addition (no exchanging)


2 1 4 1 + 1 1 2 8

3 2 6 9

5 1 8 9 + 3 1 2 8

8 3 1 711

Concrete/Visual representations could be used

alongside algorithms

2 1. 4 1 + 1. 1 2

0. 3 5

2 2. 8 8

5 1. 8 9 + 3. 1 2 8

5 5. 0 1 811


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 2 3 4 5 6 7 8 9

U 1/10 1/100

0.1

0.11 0.01

?0.7 0.3


Bar Model

0 0.1 0.2 0.3 0.4 0.5


Addition with decimals up to three decimal places including in different contexts e.g. money and measures

Cuisenaire

?

Read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit

Count forwards or backwards in steps of powers of 10 for any given number up to 1 000 000

Interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero

Round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000

Begin to round decimal numbers to the nearest whole number

Solve number problems and practical problems that involve all of the above

Practise mental calculations with increasingly large numbers

Read Roman numerals to 1000 (M) and recognise years written in Roman numerals.

Add and subtract whole numbers with more than 4 digits, and decimals up to 3 decimal places, including using formal written methods (columnar addition and subtraction)

Add and subtract numbers mentally with increasingly large numbers

Use rounding and estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.

N.B. ENSURE CHILDREN HAVE THE OPPORTUNITY TO ADD MORE THAN TWO NUMBERS INCLUDING DECIMALS, WITH DIFFERING NUMBERS OF DIGITS

End of Year Expectations Teacher Modelling/Children’s Recording

Addition

Fluency

Year 6

Column addition (no exchanging)


2 1 4 1 + 1 1 2 8

3 2 6 9

5 1 8 9 + 3 1 2 8

8 3 1 711

Concrete/Visual representations could be used

alongside algorithms

2 1. 4 1 + 1. 1 2

0. 3 5

2 2. 8 8

5 1. 8 9 + 3. 1 2 8

5 5. 0 1 811


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 2 3 4 5 6 7 8 9

U 1/10 1/100

0.1

0.11 0.01

?0.7 0.3


Bar Model

0 0.1 0.2 0.3 0.4 0.5


Addition with decimals up to three decimal places including in different contexts e.g. money and measures

Cuisenaire

?

Read, write, order and compare numbers up to 0 000 000 and determine the value of each digit

Count in tens and hundreds increasing fluency of order and place value

Round any whole number to a required degree of accuracy

Round decimal numbers to the nearest whole number and to one or two decimal places

Use negative numbers in context, and calculate intervals across zero

Add numbers mentally (HTU+ HTU) or (TH.th + TU.th)

Solve number and practical problems that involve all of the above.

Use number-bond knowledge to derive decimal number-bonds (0.6 + 0.4 = 1) or (0.63 + 0.37 = 1)

Perform mental calculations, including with mixed operations and large numbers

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

Solve problems involving addition, subtraction, multiplication and division using their knowledge of the order of operations to carry out calculations involving the four operations

Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.

Add numbers with more than four-digits and decimals up to three places (formal written column method)

N.B. ENSURE CHILDREN HAVE THE OPPORTUNITY TO ADD MORE THAN TWO NUMBERS, INCLUDING DECIMALS, WITH DIFFERING NUMBERS OF DIGITS

16

PartitioningTake away

… how many left?How many are not?How many do not?

Structures of Subtraction (Haylock and Cockburn 2008)Children should experience problems with all the different subtraction structures in a

range of practical and relevant contexts e.g. money and measurement

ReductionStart at and reduce by

Count back byGo down by

ComparisonWhat is the difference?

How many more?How many less (fewer)?

How much greater?How much smaller?

Inverse-of-additionWhat must be added?

How many (much) more needed?

There are ten pegs on the hanger –

how many are covered?

1 2 3 4 5 6 7 8 9 10

-1 -1

‘two more than three is five or two less than

five is three’


Addition





Year R

Fluency

Count backward in ones from 5 (five little ducks)

etc,

Count backward in ones from 10.

Count backward in ones from any number less

than 20.

Find one less than a number

Read digits up to 20

Match written numbers to number of objects

Order concurrent numbers upto 20 from

largest to smallest

Recognise and use the -symbol

Children count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number.

Using quantities and objects, they add and subtract twosingle-digit numbers and count on or back to find the answer.

They solve problems, including doubling, halving and sharing.


Understand what a number looks like.. Eg what is 6? 6 bears. 6 pencils. 6 children etc.


encourage counting back in ones and then groups.

‘’one more than three is

four. One less than

four is three’

1 2 3 4 5 876 90


10

End of Year Expectations Children’s Recording

Subtraction

Children may begin recording pictorially progressing to recording

number sentences alongside

Children could use printed Numicon icons and stick these in,

again progressing to recording number sentences alongside

Year 1

Fluency

Example

0 1 2 3 4 5 6 7 8 9 10

-1 -1

Use practical resources such as bears, counters, cubes and number lines/hundred grids and progress to a resource such as Numicon to encourage counting

back in groups rather than ones

‘two less than five is

three’

1 2 3 4 5 876 90 10

0 1 2 3 4 5 6 7 8 9 10



5 - 3Read, 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 number problems such as 7 = ___ – 9.

Compare quantities to say how many less and/or how many more

Understand subtraction as taking away What is … less than …?)

Represent and use number bonds to 5

Represent and use number bonds to ten

Use numberbonds to ten to derive subtraction

facts

Count backwards (including crossing 100)

any given number

Switch count between ones and tens e.g. 33,

32, 31, 30, 20, 10

Represent and use number bonds up to 20 (establish addition and subtraction as related

operations)

Count, read and write numerals to 100

Read and write numbers to 20 in numerals or

words

Find one less than a number

Find ten less than a number

identify and represent numbers using objects

and pictorial representations

including the number line, and use the

language of: equal to, more than, less than (fewer), most, least

Count back in multiples of 2s, 5s and 10s

starting on multiples to highlight pattern

Children should use numberlines to count back/take away by

counting back in ones

End of Year Expectations Children’s Recording

Subtraction

Fluency

Year 2

Children apply, develop and secure their understanding of place value and

begin to record using jottings and number

sentences

16 - 3

no exchanging

exchangingexchange ten for ten ones

26 - 8

10


Finding the difference

Children should use concrete materials and

pictorial representations, and use numbers

in different contexts e,g, money and

measures, Encourage children to partition

numbers rather than counting in

ones.

20 2

0 10 20 30 40 50 60 70 80 90Numbered and partially numbered number lines


10 - 4

7 ?10

Bar Model

Cuisenaire

?

Practise addition and subtraction facts to 20 Show increasing fluency in deriving subtraction facts for numbers up to 10 and then up to 20

Subtract numbers mentally by counting back TU - U or U - U or TU - TU(crossing 10’s barrier)

Count backward in steps of 2, 3, and 5 from 0, and in tens from any number.

Recognise the place value of each digit in a two-digit number (tens, ones)

Identify, represent and estimate numbers using different representations, including the number line

Compare and order numbers from 0 up to 100; use <, > and = signs

Read and write numbers to at least 100 in numerals and in words

Use place value and number facts to solve problems.

Use known facts to 20 to derive new facts e.g. 10 - 7 /100 – 70

Use knowledge to derive and use subtraction number facts up to 100 (multiples of 10)

Solve problems with addition and subtraction: - using concrete objects and pictorial representations, including those involving numbers, quantities , measures, money and real life contexts -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

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 solve missing number problems.

Understand subtraction as taking away and finding the difference

Be able to partition numbers in different ways

Children should use numberlines to count

back or on and find the difference, developing into subtracting bigger

chunks than one


Teacher Modelling/ Children’s Recording

Year 3

Fluency

Column subtraction (no exchange)

1 0 0 4 0 8 1 0 0 2 0 1

0 + 2 0 + 7 = 2 7

1 4 8 - 1 2 1 2 7

6 0 8

2 0 3

4 0 + 5 = 45

no exchange

Children SHOULD use manipulatives alongside algorithms to transition between practical and abstract

with exchange

6 0 3

2 0 8

3 0 + 5 = 3 5

10 +50

68 - 23 63 - 28

148 -121



110100

1

1

100 10 3

100

7?30

0 10 20 30 40 50

Subtraction

Ensure children can solve calculations where zero is a place holder

Bar Model

Cuisenaire

?

Consolidate confidence with

numberlines, finding difference by

counting on in larger chunks

Use dienes alongside to begin to develop standard methods of subtraction (starting with partitioning to complete the

expanded method)

Count back in ones, tens and hundreds maintaining fluency through varied and frequent practice

Switch count between hundreds, tens and ones e.g 500, 400, 300, 290, 280, 270, 269, 268, 267

Find 10 or 100 less than a given number with up to three digits Recognise the place value of each digit in a three-digit number (hundreds, tens, ones)

Mentally subtract HTU - ones, HTU - tens, HTU - hundreds

Perform mental calculations with two-digit numbers

Compare and order numbers up to 1000


Read and write numbers up to 1000 in numerals and in words

Solve number problems and practical problems involving these ideas.

Add and subtract numbers mentally, including: a three-digit number and ones a three-digit number and tens a three-digit number and hundreds


Add and subtract numbers with up to three digits, using formal written methods of columnar subtraction

Estimate the answer to a calculation and use inverse operations to check answers

Solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.


Teacher Modelling/ Children’s Recording Fluency

Year 4

Column subtraction (no exchange)

1 0 0 4 0 8 1 0 0 2 0 1

0 + 2 0 + 7 = 2 7

Column subtraction (with exchange)

1 4 8 - 1 2 1 2 7

6 0 8

2 0 3

4 0 + 5 = 45

no exchange

Children SHOULD use manipulatives alongside algorithms to transition between practical and abstract

with exchange

6 0 3

2 0 8

3 0 + 5 = 3 5

10 +50

68 - 23 63 - 28

148 -121

723 -367

7 2 3 - 3 6 7 3 5 6

1116



110100

1

1

200 30 3

100

7?30

0 10 20 30 40 50

Subtraction


Bar Model

Cuisenaire

723 -317 7 2 3 - 3 1 7

1

4 0 6

1

?

100100

10

10

Count back in multiples of 6, 7, 9, 25 and 1000

Find 1000 less than a given number

Count backwards through zero to include negative numbers

Subtract numbers mentally by partitioning (TU - TU)

Use adjusting to subtract mentally (45 - 19)

Continue to practise mental subtraction calculations with increasingly large numbers to aid fluency

Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones)

Order (in descending order) and compare numbers beyond 1000


Round any number to the nearest 10, 100 or 1000

Solve number and practical problems that involve all of the above and with increasingly large positive numbers

Read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value.

Subtract numbers with up to 4 digits using the formal written methods of columnar subtraction where appropriate

Estimate and use inverse operations to check answers to a calculation

Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and justifying why .

Understand subtraction as the inverse of addition

End of Year Expectations Teacher Modelling/ Children’s Recording

Year 5

Fluency

Column subtraction (no exchanging)

Column subtraction (with exchanging)

1 3 5 4 8 - 1 2 1 2 8

1 4 2 0 1 3 4 2 3 - 1 2 6 7 8 7 4 5

11132 1

1 . 4 8 - 1 . 2 10. 2 7

7 . 2 3 - 3 . 6 7 3 . 5 6

1116

Children might use manipulatives alongside algorithms



Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together.Subtraction

Subtraction with decimals up to three decimal places including in different contexts e.g. money

and measures

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 2 3 4 5 6 7 8 9

U 1/10 1/100

0.1

0.11 0.01

1

? 0.3

Bar Model

0 0.1 0.2 0.3 0.4 0.5


Cuisenaire


?

Pupils to understand the value of numberline subtraction

for contextual problems

(time/money/ measures etc)

Read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit

Count backwards in steps of powers of 10 for any given number up to 1 000 000

Subtract numbers mentally by partitioning (HTU- TU or HTU - HTU)

Use adjusting to subtract mentally (45 - 19)

Interpret negative numbers in context, count backwards with positive and negative whole numbers, including through zero

Round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000

Begin to round decimal numbers to the nearest whole number

Practise mental calculations with increasingly large numbers

Solve number problems and practical problems that involve all of the above

Read Roman numerals to 1000 (M) and recognise years written in Roman numerals.

Subtract whole numbers with more than 4 digits and decimals, including using formal written methods (columnar subtraction)

Subtract numbers mentally with increasingly large numbers

Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why. (this may include number lines)

End of Year Expectations Teacher Modelling/ Children’s Recording Fluency

Year 6



1 3 5 4 8 - 1 2 1 2 8

1 4 2 0 1 3 4 2 3 - 1 2 6 7 8 7 4 5

11132 1

1 . 4 8 - 1 . 2 10. 2 7

7 . 2 3 - 3 . 6 7 3 . 5 6

1116

Children might use manipulatives alongside algorithms



Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together.Subtraction

Subtraction with decimals up to three decimal places including in different contexts e.g. money and measures

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 2 3 4 5 6 7 8 9

U 1/10 1/100

0.1

0.11 0.01

1

? 0.3

Bar Model

0 0.1 0.2 0.3 0.4 0.5


Cuisenaire


?

Pupils to understand the value of numberline subtraction

for contextual problems

(time/money/ measures etc)

Read, write, order and compare numbers up to 0 000 000 and determine the value of each digit

Count back in tens and hundreds increasing fluency of order and place value

Round any whole number to a required degree of accuracy

Round decimal numbers to the nearest whole number and to one or two decimal places

Use negative numbers in context, and calculate intervals across zero

Subtract numbers mentally (HTU - HTU) or (TH.th - TU.th)

Solve number and practical problems that involve all of the above.

Use number-bond knowledge to derive decimal number-bonds (1 - 0.4 = 0.6) or (1 - 0.37 = 0.63)

1 . 6 8 7 0 - 1 . 2 1 4 6

Column subtraction (with differing numbers of digits-adding place

holder zero)

Perform mental calculations, including with mixed operations and large numbers

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why (this could include number lines)

Solve problems involving addition, subtraction, multiplication and division using their knowledge of the order of operations to carry out calculations involving the four operations

Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy.

Subtract numbers with more than four-digits and decimals up to three places (formal written column method)

N.B. ENSURE CHILDREN HAVE THE OPPORTUNITY TO SUBTRACT DECIMALS, WITH DIFFERING NUMBERS OF DIGITS

greenfields federation littlehaven infant school northolmes junior school

Documents

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tu u

tu u

tu tu

u tu numbers

tu u numbers

thhtu x u

thhtu x tu