1. Greenfields Federation Littlehaven Infant School Northolmes Junior School

2. 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.

3. 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

4. Magnitude of Calculations Reception– 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

5. 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

6. Symbols Concrete Experiences Pictures Language 12 + 19 Active/concrete Building visual images Abstract 13 - 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.

7. Numicon bead string 1 0 1 count stick Cuisenaire place value apparatus number line number grids 100 and 200 Multilink 0.1 place value counters 1 0.1 10 100 double sided counters

8. Aggregation Union 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 Augmentation Start at and count on Increase by Go up by Commutative law Understand addition can be done in any order Start with bigger number when counting on (Explain to children that subtraction does not have this property) 67 8 9101112131415 +1 is the same as/equal to (=)

9. 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 two single-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’ 1234587690 Possible Concrete and Visual Representations 10 5 + 4

10. NC End of Year Expectations Children’s Recording Addition Fluency Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. Possible Concrete and Visual Representations 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 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 + = 1 + 2 = 3 9 + 6 6 78 910111213141516 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 groups rather than ones ‘ two more than three is five or two less than five is three’ +2 1234587690 4 + 2 two more than four 2 4 681012141618

11. NC End of Year Expectations Children’s Recording Addition Fluency Year 2 Children apply, develop and secure their understanding of place value Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. 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 Possible Concrete and Visual Representations 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. 202 05 10 1520 25 30354045 Numbered and partially numbered number lines 73 ? ? 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

12. NC End of Year Expectations Teacher Modelling/Children’s Recording Addition Fluency Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. They should be taught together. Possible Concrete and Visual Representations 10 Year 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 exchange 1 ? 7 70 1 10 100 10 3 Partially numbered and blank number lines Cuisenaire 30 Bar 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 + 2843 + 28 40 + 20 + 1 + 8 = 6940 + 20 + 3 + 8Or 41 + 20 + 8 = 6943 + 20 + 8

13. 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 recording Compact (column) 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 exchange Compact (column) recording £ 7. 8 9 + £ 6. 4 2 £ 1 4. 3 1 7 8 9 + 6 4 2 1 4 3 1 11 H T U 11 Possible Concrete and Visual Representations 1 10 100 1 1 10 3 ? 7 Bar Model 7030 0 10 2030 40 50 Partially numbered and blank number lines Add decimals in the context of money 1 0 1 4 3 + 1 2 8 2 7 1 1 Cuisenaire ? 10 100 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 Identify, represent and estimate numbers using different representations 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

14. End of Year Expectations Teacher Modelling/Children’s Recording Addition Year 5 Fluency Column addition (no exchanging) Column addition (with exchanging) 2 1 4 1 + 1 1 2 8 3 2 6 9 5 1 8 9 + 3 1 2 8 8 3 1 7 11 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 8 1 1 Possible Concrete and Visual Representations 0.010.020.030.040.050.060.070.080.09 0.10.20.30.40.50.60.70.80.9 123456789 U 1/10 1/100 0.10.01 1 0.1 1 0.01 ? 0.7 0.3 Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. Bar Model 0 0.1 0.2 0.3 0.40.5 Partially numbered and blank number lines 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

15. End of Year Expectations Teacher Modelling/Children’s Recording Addition Fluency Year 6 Column addition (no exchanging) Column addition (with exchanging) 2 1 4 1 + 1 1 2 8 3 2 6 9 5 1 8 9 + 3 1 2 8 8 3 1 7 11 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 8 1 1 Possible Concrete and Visual Representations 0.010.020.030.040.050.060.070.080.09 0.10.20.30.40.50.60.70.80.9 123456789 U 1/10 1/100 0.10.01 1 0.1 1 0.01 ? 0.7 0.3 Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. Bar Model 0 0.1 0.2 0.3 0.40.5 Partially numbered and blank number lines 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. 16 Partitioning Take 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 Reduction Start at and reduce by Count back by Go down by Comparison What is the difference? How many more? How many less (fewer)? How much greater? How much smaller? Inverse-of-addition What must be added? How many (much) more needed? There are ten pegs on the hanger – how many are covered? 123456789 10 ‘two more than three is five or two less than five is three’

17. 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 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 two single-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 back in ones and then groups. ‘ ’one more than three is four. One less than four is three’ 1234587690 Possible Concrete and Visual Representations 10

18. 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 Exam ple 0 123456789 10 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’ 1234587690 10 0123456789 Possible Concrete and Visual Representations Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. 5 - 3 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 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

19. 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 exchanging exchange ten for ten ones 26 - 8 10 Possible Concrete and Visual Representations 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 2 010 20 30405060 708090 Numbered and partially numbered number lines Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. 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

20. End of Year Expectations 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 Possible Concrete and Visual Representations Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. 1 10 100 11 10 3 100 7 ? 30 0 10 2030 4050 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 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 Children apply, develop and secure their understanding of place value and begin to record in columns 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.

21. End of Year Expectations Teacher Modelling/ Children’s RecordingFluency 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 3 5 6 1116 Possible Concrete and Visual Representations Pupils develop the concept of addition and subtraction and are enabled to use these operations flexibly. Addition and subtraction should be taught together. 1 10 100 11 200 30 3 100 7 ? 30 0 10 2030 4050 Subtraction Ensure children can solve calculations where zero is a place holder Bar Model Cuisenaire 723 -317 1 4 0 6 1 ? 100 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 Identify, represent and estimate numbers using different representations 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

22. 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 111321 1. 4 8 - 1. 2 1 0. 2 7 7. 2 3 - 3. 6 7 3. 5 6 1116 Children might use manipulatives alongside algorithms Column subtraction (no exchanging) Column subtraction (with exchanging) 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 0.010.020.030.040.050.060.070.080.09 0.10.20.30.40.50.60.70.80.9 123456789 U 1/10 1/100 0.10.01 1 0.1 1 0.01 1 ?0.3 Bar Model 0 0.1 0.20.3 0.40.5 Ensure children can solve calculations where zero is a place holder Cuisenaire Possible Concrete and Visual Representations ? 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)

23. End of Year Expectations Teacher Modelling/ Children’s RecordingFluency Year 6 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 111321 1. 4 8 - 1. 2 1 0. 2 7 7. 2 3 - 3. 6 7 3. 5 6 1116 Children might use manipulatives alongside algorithms Column subtraction (no exchanging) Column subtraction (with exchanging) 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 0.010.020.030.040.050.060.070.080.09 0.10.20.30.40.50.60.70.80.9 123456789 U 1/10 1/100 0.10.01 1 0.1 1 0.01 1 ?0.3 Bar Model 0 0.1 0.20.3 0.40.5 Ensure children can solve calculations where zero is a place holder Cuisenaire Possible Concrete and Visual Representations ? 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