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. 2

3. 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. 4 Magnitude of Calculations 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. 6 Repeated addition So many lots (sets) of so many How many (how much) altogether Per, each Structures of Multiplication (Haylock and Cockburn 2008) Children should experience problems with all the different multiplication structures in a range of practical and relevant contexts e.g. money and measurement Scaling Scaling, scale factor Doubling, trebling So many times bigger than (longer than, heavier than, and so on) So many times as much as (or as many as) Commutative law Scaling, scale factor Doubling, trebling So many times bigger than (longer than, heavier than, and so on) So many times as much as (or as many as) a x b and b x a are equal scaling with Cuisenaire I’m 3 times as tall as you. I’m 3 metres tall I’m only 1 metre tall 4 x 2 is the same as/equal to 2 x 4 4 103

7. End of Year Expectations Possible concrete and visual representation Children’s Recording Multiplication Year R Fluency Practical only e.g. link to small world Using concrete objects, pictorial representations and arrays with the support of an adult – take photographs/draw pictures – if using Numicon small icons could be stuck in two lots of two is four arrays- Numicon, Cuisenaire, counters counting in twos 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 multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. 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 Use objects to count in twos and fives Recognise dots/objects that are the same or have the same value 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 twos and fives..

8. End of Year Expectations Possible concrete and visual representation Children’s Recording Multiplication Year 1 Fluency e.g. 6, 8, 10, 12 etc Emphasise number patterns Double numbers and quantities Practical only e.g. link to small world Using concrete objects, pictorial representations and arrays with the support of an adult – take photographs/draw pictures – if using Numicon small icons could be stuck in flexible array track with cuisenaire four lots of two is eight two lots of four is eight arrays- Numicon, Cuisenaire, counters counting in twos Solve single step practical problems involving multiplication and division Use concrete objects, pictorial representations and arrays to explore grouping Double numbers and quantities Make connections between arrays, number patterns and counting in twos, fives and tens 0 2468 Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities Add using doubles Switch count between tens and ones e.g. 10, 20, 30, 31, 32, 33 … Count in multiples of 2s, 5s and 10s starting on multiples to highlight pattern recognition

9. End of Year Expectations Possible concrete and visual representation Children’s Recording Multiplication Fluency Year 2 Record practical work as number sentences, this may go alongside arrays/photos of physical multiplication in their books/whiteboards Count in twos, threes, fives from zero and tens from any number forward and begin to do it backwards e.g. 6, 8, 10, 12 etc Emphasise number patterns Introduction to multiplication tables. Practise to become fluent in multiplication facts for 2, 5 and 10 Solve multiplication problems mentally using arrays, contexts and materials to help them, Begin to use know multiplication facts to help them to find new facts, including related division facts 2 + 2 + 2 + 2 = 4 x 2 two add two add two add two add two = four lots of two 4 x 2 = 8 2 x 4 = 8 Ensure children understand that multiplication is commutative (can be done in any order) Understand multiplication as repeated addition Understand and solve problems involving arrays flexible array Calculate mathematical statements for multiplication and division within the tables and write them using symbols (×), (÷) (=) 51015 Begin to understand that multiplication and division are inverse operations 246 08 0 2468 Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. Understand and solve problems in contexts using arrays, physical objects, repeated addition or multiplication facts to solve Pupils use a variety of language to describe multiplication and division. Connect 10 times table to place value. Connect 5 times table to divisions on clock face

10. End of Year Expectations Teacher Modelling/Children’s Recording Year 3 Fluency Count from 0 in multiples of 4, 8, 50 and 100 Practise, Recall and use multiplication table facts for the 3,4 and 8 times tables Use multiples of 2, 3, 4, 5, 8, 10, 50 and 100 Connect the 2, 4 and 8 times tables using doubling Develop efficient mental methods using commutativity and multiplication facts to derive related facts e.g. 4 x 4 x 12 = 12 x 4 x 5 = 12 x 20 Multiplication – multiplication and division should be taught together– refer to structures of multiplication Expanded methods – grid and partitioning Some children MAY Progress to developing fluency in short multiplication Understand and solve problems in context including, missing number problems, measuring and scaling problems Solve problems involving multiplication including correspondence in which n objects is connected to m objects 4 103 4 x 13 ‘four lots of thirteen’ x 10 3 4 40 12 40 + 12 = 52 1 3 x 4 5 2 1 Start with digits that are below five so children can practise method without encountering difficulty with multiplication tables Children must use manipulatives alongside algorithms Possible concrete and visual representation Cuisenaire to represent scaling arrays 4 x 13 bar models ? 20 ? 10 1 11 1 11 1 1 1 11 1 Statue is 3 times as tall: 3 metres I am 1 metre tall flexible array place value counters Pupils continue to count in ones, tens and hundreds write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers Use mental methods including partitioning, progressing to reliable (more formal) written methods Pupils develop reliable written methods for multiplication, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication. Multiply and divide whole numbers by ten and hundred(not decimals) Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one- digit numbers or quantities by 10 Multiply by 20 by x 2 and x 10, understand that this works as 20 is two lots of ten 18 X 5 = 10 X 5 = 50 8 X 5 = 40 90 Expanded/ ladder method of multiplication

11. End of Year Expectations Teacher Modelling/Children’s Recording Fluency Year 4 Multiplication – multiplication and division should be taught together– refer to structures of multiplication Count in multiples of 6, 7, 9, 25 and 1000 Recall and use multiplication facts and related division facts up to 12 x 12 with increasing fluency Derive multiplication facts with up to three-digits (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6). Recognise and use factor pairs and commutativity in mental calculations Use the distributive law (multiplying a number by a group of numbers added together is the same as doing each multiplication separately Eg: 3 × (2 + 4) = 3×2 + 3×4) Combine knowledge of number facts and rules of arithmetic to solve mental and written calculations e.g. 2 x 6 x 5 = 10 x 6 Possible concrete and visual representation Multiplying three numbers together eg 2 x 3 x 4 – associative law –(2 x 3) x 4 = 2 x (3 x 4) Multiplying by 0 and by 1 Use mental methods including partitioning, ladder and grid method progressing to reliable (more formal) written methods For those who are confident, develop fluency in short multiplication using formal written layout (2 and 3 digit numbers by a 1 digit number). Solve problems involving multiplication including using the distributive law to multiply a 2 digit number by a 1 digit number (24 x 7 = 20 x 7 + 4 x7), integer scaling problems (I am two times bigger/I am five times bigger)and harder correspondence problems such as n objects are connected to m objects Solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or three cakes shared equally between 10 children. Cuisenaire to represent scaling 4 x 13 bar models ? 20 ? 10 1 11 1 11 1 1 1 11 1 Statue is 3 times as tall: 3 metres I am 1 metre tall flexible array place value counters They begin to extend their knowledge of the number system to include the decimal numbers and fractions that they have met so far. Multiply and divide whole numbers by ten and a hundred (including decimal s) Count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten. Expanded methods – grid, ladder and partitioning 4 103 4 x 13 ‘four lots of thirteen’ x 10 3 4 40 12 40 + 12 = 52 1 3 x 4 5 2 1 3 3 x 4 5 3 2 1 11 18 X 5 = 10 X 5 = 50 8 X 5 = 40 90 MOST children SHOULD progress to developing fluency in short multiplication Higher ability children could look at using this method involving decimals Children must use manipulatives alongside algorithms

12. End of Year ExpectationsTeacher Modelling/Children’s Recording Year 5 Fluency Short multiplication Count forwards in steps of powers of 10 from any given number up to 1 000 000 Apply all multiplication tables frequently. Commit them to memory and use them confidently to make larger calculations Multiply numbers mentally drawing upon known facts Practise and develop knowledge of factors, multiples, square numbers and prime numbers Multiply whole numbers and those involving decimals by 10, 100 & 1000 Multiplication - multiplication and division should be taught together– refer to structures of multiplication 1 3 2 4 x 6 7 9 4 4 2 11 multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers Identify multiples and factors including finding all factor pairs of a number, and common factors of two numbers Solve problems involving all four operations where larger numbers are used, using knowledge of multiples, factors, squares and cubes. Multiply whole numbers and those involving decimals by 10, 100 & 1000 Understand and use multiplication and division as inverses including in problems involving missing numbers and balancing equations Solve problems involving multiplication and division including scaling by simple fractions Know and use the vocabulary of prime numbers, prime factors and composite (non-prime), recall prime numbers to 19 and work out if any number up to 100 is prime recognise and use square numbers and cube numbers, and the notation for squared ( 2 ) and cubed ( 3 ) Long multiplication 3. 2 4 x 6 1 9. 4 4 21 3. 2 4 x 2 6 1 9. 4 4 6 4. 8 0 2 1 8 4. 2 4 1 Children might use manipulatives alongside algorithms 1 3 2 4 x 2 6 7 9 4 4 2 6 4 8 0 1 3 4 4 2 4 21 1 11 1 Multiply decimals with up to three decimal places by a one digit number Possible concrete and visual representation Cuisenaire to represent scaling arrays bar models ? 0.2 ? 0.1 0.01 Statue is 3 times as tall: 3 metres I am 1 metre tall place value counters flexible array 4 x 13 4 x 23 Continue to use number in context, including measurement. Extend and apply their understanding of the number system to the decimal numbers and fractions that they have met so far. Recognise and describe linear number sequences, including those involving fractions and decimals, and find the term-to-term rule.

13. End of Year ExpectationsTeacher Modelling/Children’s RecordingFluency Year 6 Short multiplication Undertake mental calculations with increasingly large numbers and more complex calculations Continue to use all multiplication tables to calculate mathematical statements in order to maintain fluency Practise and develop knowledge of factors, multiples, square numbers and prime numbers Multiplication - multiplication and division should be taught together– refer to structures of multiplication 1 3 2 4 x 6 7 9 4 4 2 11 Long multiplication 3. 2 4 x 6 1 9. 4 4 21 3. 2 4 x 2 6 1 9. 4 4 6 4. 8 0 2 1 8 4. 2 4 1 Children might use manipulatives alongside algorithms 1 3 2 4 x 2 6 7 9 4 4 2 6 4 8 0 1 3 4 4 2 4 21 1 11 1 Possible concrete and visual representation Cuisenaire to represent scaling arrays bar models ? 0.2 ? 0.1 0.01 Multiply numbers up to 4-digit x TU using formal methods of short and long multiplication Multiply numbers with up to two decimal places x whole number Solve contextual problems involving all four operations, deciding which operations and methods to use and why Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Statue is 3 times as tall: 3 metres I am 1 metre tall place value counters flexible array 4 x 13 4 x 23 Multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, ¼ x ½ = 1/8) Use common factors to simplify fractions; use common multiples to express fractions in the same denomination Identify common factors, common multiples and prime numbers Use their knowledge of the order of operations to carry out calculations involving the four operations Recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. Multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.

14. Structures for Division (Haylock and Cockburn 2008) Children should experience problems with the different division structures in a range of practical and relevant contexts e.g. money and measurement Equal-sharing Sharing equally between How many (much) each? Inverse of multiplication (Grouping) So many lots (sets/groups) of so many Share equally in to groups of … Divide twelve into equal groups of four Make 12 Overlay groups of four = 3 Ratio structure comparison inverse of scaling structure of multiplication scale factor (decrease) Barney earns three times more than Fred. If Barney earns £900 how much does Fred earn? Jo’s journey to school is three times as long as Ella’s. If Jo walks to school in 30 minutes how long does it take Ella?

15. End of Year Expectations Possible concrete and visual representation Children’s Recording Multiplication Year R Fluency Practical only e.g. link to small world Using concrete objects, pictorial representations and arrays with the support of an adult – take photographs/draw pictures – if using Numicon small icons could be stuck in arrays- Numicon, Cuisenaire, counters counting in twos 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 multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. 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 Use objects to count in twos and fives Recognise dots/objects that are the same or have the same value 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 to share objects between a set number of people. Use objects and put them into groups of 5 etc Group numicon into twos to see there are four groups

16. End of Year Expectations Possible concrete and visual representation Teacher Modelling/Children’s Recording Division Year 1 Fluency Record as number sentences using ÷ and = Count in twos, fives and tens from different multiples e.g. 6, 8, 10, 12 etc Emphasise patterns Find simple fractions eg half and quarter, of objects, numbers and quantities Solve single step practical problems involving division Use concrete objects, pictorial representations and arrays Understand division as grouping and sharing flexible array counting in groups of twos Using concrete objects, pictorial representations and arrays with the support of an adult – take photographs/draw pictures – if using Numicon small icons could be stuck in Eight can be divided into four equal groups of two or two equal groups of four 8 ÷ 4 Eight divided into four equal groups = two in each group Use the language of ‘sharing equally between’ Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. Numicon and counter arrays doubling Cuisenaire four lots of two two lots of four Eight can be divided into four equal groups of two or two equal groups of four straw bundles bar models ? 8 ??? ? ? ?? Read and write numbers from 1 to 20 in numerals and words. Recognise, find and name a half as one of two equal parts of an object, shape or quantity Recognise, find and name a quarter as one of four equal parts of an object, shape or quantity. Connect halves and quarters to equal sharing and quantities of sets Half and quarters of shapes and quantities Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities

17. End of Year Expectations Possible concrete and visual representation Teacher Modelling/Children’s Recording Division Fluency Year 2 Record as number sentences using ÷ and = Solve division problems involving grouping and sharing Use concrete objects, pictorial representations Find halves and then quarters flexible array counting in groups of twos Using concrete objects, pictorial representations and arrays with the support of an adult – take photographs/draw pictures – if using Numicon small icons could be stuck in Eight can be divided into four equal groups of two or two equal groups of four 8 ÷ 4 Eight divided into four equal groups = two in each group Pupils develop the concept of multiplication and division and are enabled to use these operations flexibly. Multiplication and division should be taught together. Numicon and counter arrays doubling Cuisenaire four lots of two two lots of four Eight can be divided into four equal groups of two or two equal groups of four straw bundles bar models ? 8 ??? ? ? ?? Count back in twos, threes, fives TO zero and tens e.g. 12, 10, 8, 6 etc Emphasise number patterns Introduction to multiplication tables. Practise to become fluent in multiplication facts for 2, 5 and 10 – make links between multiplication facts and known division facts Begin to use know multiplication facts to help them to find new facts, including related division facts Pupils use a variety of language to describe multiplication and division. Connect 10 times table to place value. Connect 5 times table to divisions on clock face Understand division as grouping or sharing Calculate mathematical statements for multiplication and division within the tables and write them using symbols (×), (÷) (=) Understand and solve single steps practical problems involving division Begin to understand that multiplication and division are inverse operations and begin to relate these to fractions and measures (for example, 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (for example, 4 × 5 = 20 and 20 ÷ 5 = 4). Show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot Work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete quantities and to arrays Recognise, find, name and write fractions 1/3, ¼,2/4 and ¾ of a length, shape, set of objects or quantity Write simple fractions for example, 2 1 of 6 = 3 and recognise the equivalence of 2/4 and 1/2. Fractions of quantities 1/3 of 9 = 3

18. End of Year ExpectationsTeacher Modelling/Children’s Recording Year 3 Fluency Recognise, find and name ½ and ¼ of an object, shape or quantity Understand the link between unit fractions and division Repeated subtraction - chunking Short division- no remainders Connect 1/10 to division by 10 Count in tenths 1 4 0 4 5 6 0 560 ÷ 4 1 0306090120150180 18 15 12 9 6 3 0 95 ÷ 5 = 19 95 - 50 ( 10 x 5 ) 45 - 25 ( 5 x 5 ) 20 - 20 ( 4 x 5 ) 0 Fact Box 2 x 5 = 10 5 x 5 = 25 10 x 5 = 50 Write and calculate mathematical statements for division using what is known Use division facts to derive related division facts e.g. using 6 ÷ 3 = 2 to work out 60 ÷ 3 = 20 Children should use manipulatives alongside algorithms Division - multiplication and division should be taught together– refer to structures of division Ensure children see/understand the link between grouping on a number line either horizontally or vertically recording for chunking (NOT ALL CHILDREN will use vertical chunking) arrays bar models ? 80 ??? ÷ 88 ÷ 4 Possible concrete and visual representation 321 ÷ 3 ÷ 1 2 1 3 3 6 3 Cuisenaire to represent scaling Statue is 3 metres I am 3 times smaller ÷ 2 2 4 10 11 1 1 1 1 1 1 ? ??? 10 lots of 3 Use a numberline to see how many ‘lots’ of a number fit into another number (develop to 10 lots/5 lots jumps) Use place value counters to introduce bus stop method, use place value grids to support Understand and solve problems in context including, missing number problems, measuring and scaling problems write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers Use mental methods including numberlines, progressing to reliable (more formal) written methods Pupils develop reliable written methods for division, starting with calculations of two-digit numbers by one- digit numbers with no remainders. Count from 0 in multiples of 4, 8, 50 and 100 and backwards Practise, recall and use multiplication and division facts for the 3,4 and 8 times tables Use multiples of 2, 3, 4, 5, 8, 10, 50 and 100 Connect the 2, 4 and 8 times tables using doubling Develop efficient mental methods using commutativity and multiplication and division facts to derive related facts e.g. 30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3 Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one- digit numbers or quantities by 10

19. End of Year ExpectationsTeacher Modelling/Children’s RecordingFluency Year 4 Use mental methods including numberlines, progressing to reliable (more formal) written methods then to become fluent in the formal written method of short division with exact answers when dividing by a one-digit number Divide one- or two-digit numbers by 10 or 100, identifying value of digits as tenths or hundredths Solve two-step problems in different contexts, choosing the appropriate operation, working with increasingly harder numbers including correspondence questions e.g. three cakes shared equally between 10 children Repeated subtraction - chunking Short division- no remainders 1 4 0 4 5 6 0 560 ÷ 4 1 0306090120150180 18 15 12 9 6 3 0 95 ÷ 5 = 19 95 - 50 ( 10 x 5 ) 45 - 25 ( 5 x 5 ) 20 - 20 ( 4 x 5 ) 0 Fact Box 2 x 5 = 10 5 x 5 = 25 10 x 5 = 50 Children should use manipulatives alongside algorithms Division - multiplication and division should be taught together– refer to structures of division Ensure children see/understand the link between grouping on a number line either horizontally or vertically recording for chunking (NOT ALL CHILDREN will use vertical chunking) arrays bar models ? 80 ??? ÷ 88 ÷ 4 Possible concrete and visual representation 321 ÷ 3 ÷ 1 2 1 3 3 6 3 Cuisenaire to represent scaling Statue is 3 metres I am 3 times smaller ÷ 2 2 4 10 11 1 1 1 1 1 1 ? ??? 10 lots of 3 Use a numberline to see how many ‘lots’ of a number fit into another number (develop to 10 lots/5 lots jumps) Use place value counters to introduce bus stop method, use place value grids to support Count in multiples of 6, 7, 9, 25 and 1000 Recall and use multiplication facts and related division facts up to 12 x 12 with increasing fluency Derive multiplication facts with up to three-digits (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6). Recognise and use factor pairs and commutativity in mental calculations Use place value, known and derived facts to divide mentally, including dividing by 1 Use the distributive law (multiplying a number by a group of numbers added together is the same as doing each multiplication separately Eg: 3 × (2 + 4) = 3×2 + 3×4) Combine knowledge of number facts and rules of arithmetic to solve mental and written calculations e.g. 2 x 6 x 5 = 10 x 6 They begin to extend their knowledge of the number system to include the decimal numbers and fractions that they have met so far. Multiply and divide whole numbers by ten and a hundred (including decimal s) Count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten. Solve problems involving division including integer scaling problems (I am two times smaller/I am five times smaller) Solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or three cakes shared equally between 10 children. Recognise and show, using diagrams, families of common equivalent fractions and understand link to multiplication and division Pupils understand the relation between fractions and multiplication and division of quantities,

20. Teacher Modelling/Children’s Recording Year 5 Fluency short division Higher ability children may be introduced to long division 2 3 r 8 2 4 5 6 0 - 4 8 8 0 - 7 2 8 remainder as a whole number 560 ÷ 4 1 4 0 4 5 6 0 1 564 ÷ 5 5 5 6 4 1 2/5 remainder as a fraction or as a decimal without remainder Children might use manipulatives alongside algorithms 1 1 2. 8 5 5 6 4. 0 1 4 Division - multiplication and division should be taught together– refer to structures of division End of Year Expectations Possible concrete and visual representation flexible arrays ÷ bar models ? 0.8 ??? 4.8 ÷ 4 4 1 0.1 1 11 Cuisenaire to represent scaling Statue is 3 metres I am 3 times smaller 4 0.8 1. 2 1 1 2 ? ? ?? Fact Box 2 x 24 = 48 3 x 24 = 72 4 x 24= 96 5 x 24 = 120 Fact boxes for long division may still be useful eg- Practise and extend the formal written method of short division: numbers up to four-digits by a one- digit number Interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding as appropriate for the context Identify multiples and factors including finding all factor pairs of a number, and common factors of two numbers Solve problems involving all four operations where larger numbers are used, using knowledge of multiples, factors, squares and cubes. Understand and use multiplication and division as inverses including in problems involving missing numbers and balancing equations Solve problems involving multiplication and division including scaling by simple fractions Know and use the vocabulary of prime numbers, prime factors and composite (non-prime), recall prime numbers to 19 and work out if any number up to 100 is prime recognise and use square numbers and cube numbers, and the notation for squared ( 2 ) and cubed ( 3 ) Divide whole and decimal numbers by 10,100 and 1000 Use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres Count backwards in steps of powers of 10 from any given number up to 1 000 000 Apply all multiplication tables frequently. Commit them to memory and use them confidently to make larger calculations Count backwards with positive and negative whole numbers through zero Practise and develop knowledge of factors, multiples, square numbers, cube numbers and prime numbers Multiply and divide numbers mentally drawing upon known facts Multiply and divide whole numbers and those involving decimals by 10, 100 & 1000 Continue to use number in context, including measurement. Extend and apply their understanding of the number system to the decimal numbers and fractions that they have met so far. Recognise and describe linear number sequences, including those involving fractions and decimals, and find the term-to-term rule. (20 x 15 ) (8 x 15 )

21. Teacher Modelling/Children’s Recording Fluency Year 6 Perform mental calculations, including with mixed operations and larger numbers short division long division 2 3 r 8 2 4 5 6 0 - 4 8 8 0 - 7 2 8 2 3 8/24 (1/3) 2 4 5 6 0 - 4 8 8 0 7 2 8 remainder as a whole number remainder as a fraction in its lowest form 560 ÷ 4 1 4 0 4 5 6 0 1 564 ÷ 5 5 5 6 4 1 2/5 remainder as a fraction or as a decimal without remainder Children might use manipulatives alongside algorithms 560 ÷ 24 1 1 2. 8 5 5 6 4. 0 1 4 Division - multiplication and division should be taught together– refer to structures of division End of Year Expectations Possible concrete and visual representation flexible arrays ÷ bar models ? 0.8 ??? 4.8 ÷ 4 4 1 0.1 1 11 Cuisenaire to represent scaling Statue is 3 metres I am 3 times smaller 2 3. 3 2 4 5 6 0. 0 - 4 8 8 0 7 2 8 0 7 2 remainder as a decimal 4 0.8 1. 2 1 1 2 ? ? ?? Divide numbers up to 4-digit bu a 2 digit using formal methods of short and long division interpret remainders as whole numbers, fractions or by rounding, as appropriate for the context Divide numbers with up to 2 decimal places by 1-digit and 2-digit whole numbers, initially in practical contexts involving money and measures Understand the relationship between unit fractions and division Recognise division calculations as the inverse of multiplication Solve problems involving division Solve contextual problems involving all four operations, deciding which operations and methods to use and why Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Identify common factors, common multiples and prime numbers Use their knowledge of the order of operations to carry out calculations involving the four operations Use common factors to simplify fractions; use common multiples to express fractions in the same denomination Undertake mental calculations with increasingly large numbers and more complex calculations Continue to use all multiplication tables and division facts to calculate mathematical statements in order to maintain fluency Practise and develop knowledge of factors, multiples, square numbers and prime numbers Recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. Multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9. Divide proper fractions by whole numbers [for example, 1/3 ÷ 2 =1/6) Associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8) DIVISION SHOULD BE LINKED TO OTHER AREAS SUCH AS PERCENTAGES, RATIO AND PROPORTION