EYFS Programme of Study Statement: Pupils Should Be Taught To: Programme of Study Statement: Pupils Should Be Taught To

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Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION  National Curriculum statements 2014 (statutory) in blue and bold Exemplification and activities taken from NCETM National Curriculum toolkit Nursery Reception Year 1 Year 2 EYFS Programme of Study statement: Pupils should be taught to: Programme of Study statement: Pupils should be taught to:

Mathematics development involves providing children with opportunities to practise and improve  solve one-step problems involving multiplication and division, by calculating the answer  calculate mathematical statements for multiplication and division within the multiplication tables and their skills in counting numbers, calculating simple addition and subtraction problems, and to using concrete objects, pictorial representations and arrays with the support of a teacher write them using the multiplication (x), division (÷) and equals signs describe shapes, spaces, and measures. Children should be able to: Children should be able to: Find missing numbers or symbols in a calculation: 4 x __ = 20, __ ÷ 10 = 3 Early Learning Goal 11: Numbers Use practical apparatus, arrays and images to help solve multiplication and division problems Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does she have altogether? Show how [Expected outcomes] Children count reliably with numbers from 1 to 20, place them in order and you worked this out. say which number is one more or one less than a given number. Using quantities and objects, they such as: NCETM Activity B add and subtract two single-digit numbers and count on or back to find the answer. They solve . Ben had 5 football stickers. His friend Tom gave him 5 more, how many does he have  BBC KS1 Starship Number Jumbler game, a simple animation where children choose the missing operation problems, including doubling, halving and sharing. altogether? sign in the calculation. . Share 12 sweets between two children. How many do they each have?  Write the calculation: give the children different pictures of groups of items or arrays. They then have to write EYFS Profile exemplification for the level of learning and development expected at the end of the . Find half of and double a number or quantity: the multiplication sentence to match the picture. 2p, 5p and 10p coins could be used for this activity. The EYFS Mathematics . 16 children went to the park at the weekend. Half that number went swimming. How many children can write down how many of each coin they have and the total amount. Using coins, the children could also write division calculations to match the images. Examples of doubling, halving and repeated addition for ‘expected’ achievement related to ELG:11 children went swimming?  show that multiplication of two numbers can be done in any order (commutative) and division of one taken from exemplification . I think of a number and halve it. I end up with 9, what was my original number? number by another cannot No examples of these aspects in the ‘expected’ exemplification for ELG 11 NCETM Actvity A - Noah’s Ark Children should be able to: Use their knowledge of triangles of numbers to show related number facts. e.g. If 6 x 2 Give the children the opportunity to count in twos, finding the total number of animals on board the Ark. As the children gain fluency counting in twos, start at different numbers and perhaps = 12 then 2 x 6 = 12 and 12 ÷ 6 = 2. changing from using concrete objects, to jumping in twos along a number line. Further uses could NCETM Activity C be to find the number of groups of two on the ark, again initially using tangible objects, then  Triangle of numbers: a good activity to use as a starter or plenary, demonstrating the commutativity of moving on to using a number line and demonstrating repeated subtraction. multiplication. It can be used to demonstrate one number fact and the children can suggest the others.  Function box: reinforcing the relationship of division being the inverse of multiplication. To build up to this online activity, a function box could be made and used as a visual resource in class. For example, model how a number goes in and doubles, but put the number back through the machine it will halve. There are lots of different ways to use the function box in class to deepen the understanding of the relationship between multiplication and division.  Class number sentence: using digit cards and x and ÷ and = cards. Get the children to show how we can take one known fact and find others using those numbers. As the children move the digits around, they will demonstrate their understanding of using and applying their tables knowledge. What numbers can we move around in a division sentence? Can they spot the relationship between multiplication and division? Ensure they really understand the concept behind this activity, by encouraging them to show you with practical Activity B - Arrays powerpoint apparatus This resource is based in ‘PowerPoint’. The teacher can set simple multiplication word problems  solve problems involving multiplication and division, using materials, arrays, repeated addition, mental for the children to solve. It is also useful for modelling arrays and how to write a multiplication methods, and multiplication and division facts, including problems in contexts sentence. Children should be able to: Use various methods and apparatus to help them solve word problems such as: Activity C - W hole class counting sessions There are 10 lollies in a bag. Charlie needs 30 lollies for his party. How many bags does he need to buy? Show how you worked this out. For this activity the children themselves are the objects to count. You can count in twos to find the NCETM Activity D number of shoes in a group, count fingers on hands in fives and number of toes in tens. To  TES word problems with differentiated questions: resources submitted by a Year 2 teacher, focusing on extend the children you can ask them to model how to write down this calculation or alter it to problem solving using multiplication. practise their division facts from the 2, 5 and 10 times tables.  BBC Class Clips: problem solving - how many chairs? Use multiplication facts to help the Chuckle Brothers Activity D - NRICH Share Bears: organise their tables ready for their guests’ arrival. A lovely investigation involving the children in division by sharing, and early introduction to the  Give the children a multiplication or division fact. Can they write a word problem to match it? Now swap concept of remainders. calculations with a partner and talk about how you would solve the problem.  recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers Children should be able to: Recognise a multiple of 2, 5 or 10 and use their knowledge of multiplication facts to find corresponding division facts. They can say which numbers are odd and which are even. e.g. 2 x 5 = 10, show me three more number facts using these numbers. Is 34 an odd number? How do you know? NCETM activity A:  Multiplication tables - pelmanism style matching cards. Find the pairs of cards showing the product and multiplication fact in a timed activity.  Counting stick: using a metre stick with ten divisions use sticky notes to mark multiples of 2, 5 or 10. Practise counting on and back from different numbers. As the children become more fluent, remove more sticky notes and see how they can recall multiples starting at different points counting on and back. BBC KS1 Bitesize: Camel Times Tables and Division Mine. These two games are a super resource for the children to practise their recall of multiplication and division facts independently Taken from: Non-statutory guidance: Development Matters 30 to 50 months A unique Taken from: Non-statutory guidance: Development Matters 40 to 60+ months A unique child Non-statutory guidance Non-statutory guidance child • Recognise some numerals of personal significance. Through grouping and sharing small quantities, pupils begin to understand: multiplication and Pupils use a variety of language to describe multiplication and division. • Uses some number names and number language spontaneously. • Recognises numerals 1 to 5. division; doubling numbers and quantities; and finding simple fractions of objects, numbers and • Uses some number names accurately in play. • Counts up to three or four objects by saying one number name for each item. quantities. • Recites numbers in order to 10. • Counts actions or objects which cannot be moved. Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication • Knows that numbers identify how many objects are in a set. • Counts objects to 10, and beginning to count beyond 10. tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 • Beginning to represent numbers using fingers, marks on paper or pictures. • Counts out up to six objects from a larger group. They make connections between arrays, number patterns, and counting in twos, fives and tens. multiplication table to the divisions on the clock face. They begin to use other multiplication facts, including using • Sometimes matches numeral and quantity correctly. Selects the correct numeral to represent 1 to 5, then 1 to 10 objects. related division facts to perform written and mental calculations. • Shows curiosity about numbers by offering comments or asking questions. • Counts an irregular arrangement of up to ten objects. • Compares two groups of objects, saying when they have the same number. • Estimates how many objects they can see and checks by counting them.

• Shows an interest in number problems. • Uses the language of ‘more’ and ‘fewer’ to compare two sets of objects. Pupils work with a range of materials and contexts in which multiplication and division to relate to grouping and • Separates a group of three or four objects in different ways, beginning to recognise that the • Finds the total number of items in two groups by counting all of them. sharing discrete and continuous quantities, and relating these to fractions and measures (e.g. 40 ÷ 2 = 20, 20 is a total is still the same. • Says the number that is one more than a given number. • Shows an interest in numerals in the environment. • Finds one more or one less from a group of up to five objects, then ten objects. half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (e.g. 4 x 5 = 20 and 20 • Shows an interest in representing numbers. • In practical activities & discussion, beginning to use the vocabulary involved in adding & ÷ 5= 4). • Realises not only objects, but anything can be counted, including steps, claps or jumps. subtracting. • Records, using marks that they can interpret and explain. • Begins to identify own mathematical problems based on own interests and fascinations. NB: In Y1 most strategies are based on counting but mental strategies are developing NB: In Y2 mental and written strategies are still largely interchangeable Multiplication and division structures: synonymous at this stage combining two or more Multiplication and division structures: synonymous at this stage combining two or more groups Multiplication structures: to recognise repetitive addition of groups of the same size; as an array; counting in Multiplication and division structures: repeated aggregation (addition) of groups of the same groups of the same size and counting to establish total ( repeated addition or repeated of the same size and counting to establish total ( repeated addition or repeated aggregation); steps of 2, 5 & 10 aggregation); Also division as equal sharing and also as grouping size and recorded as an array; begin to recognise sharing equally; also repeated addition of Division structures: recognise division as sharing equally; and, also repeated addition or subtraction of groups of Also division as uequal sharing and also as grouping groups of the same size i.e. divided into groups of …. the same size i.e. grouping Recording: teacher demonstration of calculation to match pictorial recording using numberlines Recording: Following teacher demonstration, children continue to develop pictorial recording Recording: teacher demonstration of calculation of pictorial recording. Children begin using Children begin using pictorial recording and numberlines. See children’s-mathematics link below. including an array; beginning to record on their own empty number line. Recording: Following teacher demonstration of calculation to match pictorial recording (number line & array), pictorial recording. See children’s-mathematics link below. children to continue to develop recording especially using pictures and their own number lines. They begin to use half, each VOCABULARY: double; half, halve; pair; group; divide into groups of; count out, share out; left, VOCABULARY: double, groups of, sets of, lots of; groups, share, left over, standard notation (including the symbols x ÷ and =) VOCABULARY: more, and, make, sum, total, altogether, take away, leave, left over Equipment and resources : number lines, counters, fingers, straws, Base 10 equipment, every day objects, counters, fingers, VOCABULARY: lots of, groups of; times, multiply, multiplied by; multiple of; once, twice, three times… ten times…; times as (big, long, wide… and so on); repeated addition; array; row, column; double, halve; share, share Equipment: every day objects, counters, fingers Equipment and resources: every day resources but also ‘maths’ resources such as number tracks, number lines, counters, fingers, straws organised into groups of 10 ITPs: Multiplication Array ; Grouping; equally; one each, two each, three each…; group in pairs, threes… tens; equal groups of; ¸, divide, divided by, http://www.childrens-mathematics.net/ see ‘galleries’ divided into; left, left over http://www.childrens-mathematics.net/ see ‘galleries’ http://www.foundati onyears.org.uk/wp-content/upl oads/2011/10/Numbers_and_Patter ns.pdf http://www.foundati onyears.org.uk/wp-content/upl oads/2011/10/Numbers_and_Patter ns.pdf NCETM video examples: ‘Multiple Representations’ and ‘ Reinforcing tables facts The commutative law of multiplication at KS1 Sharing and grouping Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws, squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes,

ITPs: Multiplication Array; Multiplication facts; Number dials; Grouping

NCETM video examples: ‘Multiple Representations’ and Reinforcing tables facts and The commutative law of multiplication at KS1 Sharing and grouping Also, BBC Learning Zone Class Clips

Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION  National Curriculum statements 2014 (statutory) in blue and bold Exemplification and activities taken from NCETM National Curriculum toolkit Nursery Reception Year 1 Year 2 School examples of multiplication School examples of multiplication Multiplication: example from exemplification (NCETM) Multiplication: example Example from exemplification Example from exemplification . Ben had 5 football stickers on each page. He has two pages of stickers. How many does he Example from exemplification: Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does School examples of multiplication School examples of multiplication have altogether? ARRAY and she have altogether? Show how you worked this out. How many feet have these three teddy bears got altogether? How many wheels do we need for these three lego cars?

For example, ‘practically’ and counting in 5s.

4 + 4 + 4 = 12 Model on a bead bar/string as ‘repeated’ addition i.e. 5 + 5 + 5 = 15 0 1 2 3 4 5 6 7 8 9 10 Strategy: begin to recognise repetitive addition of groups of the same size AND ‘5 three times equals 15’ so 5 x 3 = 15 or ‘five multiplied or times by three equals 15’.

Recording: teacher demonstration of appropriate pictorial recording where appropriate. Demonstrate recording on a number line. Show two groups of 5 football stickers. Model this on a bead bar/string. Vocabulary: groups, sets Strategy: begin to recognise repetitive addition of groups of the same size; counting in steps of 10 Demonstrate recording of two groups of 5 stickers on a number line. or 2 x 5 1 1 1 Equipment: every day objects, counters, fingers Also show as a rectangular ‘array’ i.e. two groups of 5 or 5 + 5 = 10 Recording: teacher demonstration of calculation to match pictorial recording using standard notation of + and =. Demonstrate on number line. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Vocabulary: double, groups of, sets of, lots of

Equipment: every day objects: e.g. cars, chairs, bears, children, fingers, gloves, toy cars, pairs of socks. Also ‘maths’ objects e.g. counters, Model/demonstrate as a rectangular ‘array’ using the cakes or beads (fold bead string) or counters (left). Also show as a rectangular ‘array’ (right) as ‘three rows of 5’ i.e. 5 x 3 =15

Use the ITP Multiplication Facts to

demonstrate these processes and also commutativity.

School examples of division School examples of division Example from division exemplification (NCETM) Example from division exemplification (NCETM) If we share out these cakes so everyone has one each, how many will be left over? If everyone Can we share out these cakes fairly? How shall we do it? . Share 12 sweets between two children. How many do they each have? Other example: How many sticks of 4 cubes can you make from 20 cubes? OR If 20 cubes are shared equally has two cakes, how many children will be able to have cakes today? If we put two cakes on each plate, how many plates do we need? between 4 people, how many cubes do they each get?

GROUPING Show 20 cubes. Divide the cubes into groups of 4 i.e. one group of 4, two groups of 4 and so on…..

Model this on a bead bar/string and demonstrate recording of one group of 4 cubes on a number line….. And 0 2 4 6 8 10 12 another group of 4 cubes etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20 divided into groups of 4 equals 5.

Strategy: begin to recognise sharing equally; also repetitive addition or subtraction of groups of Strategy: begin to recognise sharing equally; also repetitive addition of ‘groups’ of the same size 2 + 2 + 2 + 2 + 2 + 2 = 12 the same size i.e. grouping ÷4 1 1 1 1 1 Recording: teacher demonstration of appropriate pictorial recording where appropriate. Recording: teacher demonstration of calculation to match pictorial recording using standard Or ‘sharing’ model to create an array, as above. Vocabulary: groups, share notation of + and =. Demonstrate on number line. EQUAL SHARING Show 12 sweets. Share the sweets equally between two i.e. one for you and Equipment: every day objects, counters, fingers Vocabulary: groups, share, left over, half one for me – we’ve used 2 of the sweets. Model this on a bead bar/string and demonstrate Equipment: every day objects, counters, fingers recording of one group of 2 sweets on a number line….. Another one for you and one for me – 0 4 8 12 16 20 we’ve used 2 more of the sweets. Model this on a bead bar/string and demonstrate recording of

the second group of 2 sweets on a number line….. etc. EQUAL SHARING use same model as above. e.g. We have 20 cubes. Share the cubes equally between each ALSO: use as above with GROUPING e.g. We have 12 sweets. If each child has 2 sweets, how person – we’ve used 4 of the cubes. Model this on a bead bar/string and demonstrate recording of one group of 4 many children will have 2 sweets? i.e. two for you – that’s one group of two – that’s one person - cubes on a number line….. Another one for each person etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20 model on bead bar and record on a number line as above. shared equally between 4 equals 5.

Find missing numbers or symbols in a calculation:  ÷ 10 = 3 Represent e.g. using the Singapore bar:

The bar above represents the unknown quantity.

If we divide the bar into 10 equal sections and we know that each section has a value of 3. 3 3 3 3 3 3 3 3 3 3 Grouping ITP So, 3 +3 +3 +3 +3 +3 +3 +3 +3 +3 = 30 OR 3 x 10 = 30 So the value of the empty box here  ÷ 10 = 3 must be 30 because 30 ÷ 10 = 3 OR as above but divided into GROUPS of 10.

Opportunities for links with other mathematics domains: Opportunities for links with other mathematics domains: Cross-curricular and real life connections Cross-curricular and real life connections Learners will encounter multiplication and division in: Learners will encounter multiplication and division in: Money - when shopping and recognising prices of items, ordering items by price, finding Money – shopping: finding quantities in multiple purchases, sales prices, sharing costs. quantities in multiple purchases, sales prices, sharing costs. Measurement - calculating area and perimeter, finding journey distances, reading and calculating scales, adjusting Measurement - calculating area and perimeter, finding journey distances, reading and calculating recipe quantities. scales, adjusting recipe quantities. Data – interpreting and evaluating data, calculating amounts from pie charts and pictograms Data - interpreting and evaluating data, calculating amounts from pie charts and pictograms.

Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION National Curriculum statements 2014 (statutory) in blue and bold Exemplification and activities taken from NCETM National Curriculum toolkit Year 3 Year 4 Year 5 Year 6 Programme of Study Pupils should be taught to: Programme of Study Pupils should be taught to: Programme of Study Pupils should be taught to: Programme of Study Pupils should be taught to:  recall and use multiplication and division facts for the 3,4 and 8 times tables.  recall multiplication and division facts for multiplication tables up to 12 × 12  identify multiples and factors, including finding all factor pairs of a number, and common factors of 2  multiply multi-digit numbers up to 4 digits by a two-digit whole number using the . multiply seven by three; what is four multiplied by nine? Etc. Children should be able to: numbers formal written method of long multiplication know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers . Circle three numbers that add to make a multiple of 4: 11 12 13 14 15 16 17 18 19 . Pupils continue to practise recalling and using multiplication tables and related division facts to aid  . Look at long-multiplication calculations containing errors, identify the errors and fluency. e.g.  establish whether a number up to 100 is prime and recall prime numbers up to 19 . Leila puts 4 seeds in each of her pots. She uses 6 pots and has 1 seed left over. How many seeds determine how they should be corrected. o One orange costs nineteen pence. How much will three oranges cost? . Use the vocabulary factor, multiple and product. They identify all the factors of a given number; eg, the factors did she start with?  divide numbers up to 4 digits by a two-digit whole number using the formal o What is twenty-one multiplied by nine? of 20 are 1, 2, 4, 5, 10 and 20. They answer questions such as: . At Christmas, there are 49 chocolates in a tin and Tim shares them between himself and 7 other o How many twos are there in four hundred and forty? written method of long division, and interpret remainders as whole number o Find some numbers that have a factor of 4 and a factor of 5. What do you notice? remainders, fractions, or by rounding, as appropriate for the context members of the family. How many chocolates will each person get?  use place value, known and derived facts to multiply and divide mentally, including: o My age is a multiple of 8. Next year my age will be a multiple of 7. How old am I? . Every day a machine makes 100 000 paper clips, which go into boxes. A full box has write and calculate mathematical statements for multiplication and division using the multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers  . They recognise that numbers with only two factors are prime numbers and can apply their knowledge of multiplication tables that they know, including for two-digit numbers times one-digit . Children should be able to: 120 paper clips. How many full boxes can be made from 100 000 paper clips? multiples and tests of divisibility to identify the prime numbers less than 100. They explain that 73 children can numbers, using mental and progressing to formal written methods. . Pupils practise mental methods and extend this to three-digit numbers to derive facts, for example . Each paper clip is made from 9.2 centimetres of wire. What is the greatest number of only be organised as 1 group of 73 or 73 groups of 1, whereas 44 children could be organised as 1 group of 44, . One orange costs nineteen pence. How much will three oranges cost? 200 × 3 = 600 into 600 ÷ 3 = 200. e.g. paper clips that can be made from 10 metres of wire? o Divide thirty-one point five by ten. 2 groups of 22, 4 groups of 11, 11 groups of 4, 22 groups of 2 or 44 groups of 1. They explore the pattern of . A DJ has two different sized storage boxes for her CDs. Small boxes hold 15 CDs. . Mark drives 19 miles to work every day and 19 miles back. He does this on Mondays, Tuesdays, o Ten times a number is eighty-six. What is the number? Wednesdays, Thursdays and Fridays. How many miles does he travel to work and back in one primes on a 100-square, explaining why there’ll never be a prime number in the 10th column & the 4th column. Large boxes hold 28 CDs. The DJ has 411 CDs. How could the DJ pack her CDs?  recognise and use factor pairs and commutativity in mental calculations week?  multiply and divide numbers mentally, drawing upon known facts . Solve problems such as: Printing charges for a book are 3p per page and 75p for the Children should be able to: . Rehearse multiplication facts and use these to derive division facts, to find factors of two-digit numbers and to cover. I paid £4.35 to get this book printed. How many pages are there in the book?  solve problems, including missing number problems, involving multiplication and division, . Pupils write statements about the equality of expressions (e.g. use the distributive law 39 × 7 = 30 × including integer scaling problems and correspondence problems in which n objects are 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number multiply multiples of 10 and 100, e.g. 40 × 50. They use and discuss mental strategies for special cases of Write down the calculations that you did. Seeds are £1.45 for a packet. I have £10 to connected to m objects. facts and rules of arithmetic to solve mental and written calculations e.g. 2 x 6 x 5 = 10 x 6. harder types of calculations, for example to work out 274 + 96,< 8006 – 2993, 35 × 11, 72 ÷ 3, 50 × 900. They spend on seeds. What is the greatest number of packets I can buy? . Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she . e.g. Understand and use when appropriate the principles (but not the names) of the commutative, use factors to work out a calculation such as 16 × 6 by thinking of it as 16 × 2 × 3. They record their methods  solve problems involving multiplication and division have to buy? associative and distributive laws as they apply to multiplication: using diagrams (such as number lines) or jottings and explain their methods to each other. They compare  use estimation to check answers to calculations and determine, in the context of . Example of commutative law 8 × 15 = 15 × 8 alternative methods for the same calculation & discuss any merits &disadvantages. a problem, levels of accuracy . Example of associative law 6 × 15 = 6 × (5 × 3) = (6 × 5) × 3 = 30 × 3 = 90 NCETM ACTIVITES Children should be able to: . Example of distributive law 18 × 5 = (10 + 8) × 5 = (10 × 5) + (8 × 5) = 50 + 40 = 90  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 . Give the best approximation to work out 4.4 × 18.6 and explain why. Answer  multiply two-digit and three-digit numbers by a one-digit number using formal written layout questions such as: roughly, what answer do you expect to get? How did you arrive at . Develop and refine written methods for multiplication. They move from expanded layouts (such as the grid that estimate? Do you expect your answer to be greater or less than your estimate? method) towards a compact layout for HTU × U and TU × TU calculations. They suggest what they expect the  solve problems involving multiplying and adding, including using the distributive law to Why? multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence approximate answer to be before starting a calculation and use this to check that their answer sounds sensible. problems such as n objects are connected to m objects For example, 56 × 27 is approximately 60 × 30 = 1800.  perform mental calculations, including with mixed operations and large numbers Use mental strategies to calculate in their heads, using jottings and/or diagrams where Children should be able to:  multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000 . appropriate. For example, to calculate 24 × 15, they multiply 24 × 10 and then halve . Pupils solve two-step problems in contexts, choosing the appropriate operation, working with . Recall quickly multiplication facts up to 10 × 10 and use them to multiply pairs of multiples of 10 and 100. They increasingly harder numbers. This should include correspondence questions such as the numbers this to get 24 × 5, adding these two results together. They record their method as (24 e.g. should be able to answer problems such as: the product is 400. At least one of the numbers is a multiple of 10. of choices of a meal on a menu, or three cakes shared equally between 10 children. × 10) + (24 × 5). Alternatively, they work out 24 × 5 = 120 (half of 24 × 10), then o 185 people go to the school concert. They pay £l.35 each. How much ticket money is What two numbers could have been multiplied together? Are there any other possibilities? multiply 120 by 3 to get 360. collected?  recognise and use square numbers and cube numbers, and the notation for squared (²) & cubed (³) o Programmes cost 15p each. Selling programmes raises £12.30. How many programmes are identify common factors, common multiples and prime numbers . solve problems involving multiplication and division, including using their knowledge of factors and multiples,  sold? squares and cubes . Children should be able to answer questions such as: o How can you use factors to multiply 17 by 12? NCETM ACTIVITES . use knowledge of multiplication facts to derive quickly squares of numbers to 12 × 12 & the corresponding squares of multiples of 10. They should be able to answer problems such as: o Start from a two-digit number with at least six factors, e.g. 72. How many o tell me how to work out the area of the cardboard with dimensions 30cm by 30cm different multiplication and division facts can you make using what you know about 72? What facts involving decimals can you derive? o find two square numbers that total 45 o What if you started with 7.2? What about 0.72?  divide numbers up to 4 digits by a one-digit number using the formal written method of short division o and interpret remainders appropriately for the context Which three prime numbers multiply to make 231? o Extend written methods for division to include HTU ÷ U, including calculations with remainders. They suggest what use their knowledge of the order of operations to carry out calculations involving the four operations they expect the approximate answer to be before starting a calculation and use this to check that their answer sounds sensible. They increase the efficiency of the methods that they are using. For example: 196 ÷ 6 is . Children should be able to find answers to calculations such as 5.6 ⬜ = 0.7 or 3 × 0.6, approximately 200 ÷ 5 = 40 drawing on their knowledge of number facts and understanding of place value. They 4 2 should be able to approximate, use inverses and apply tests of divisibility to check 32 r 4 or or 6 3 their results. 6 196 . Children should know the square numbers up to 12 × 12 and derive the corresponding Children know that, depending on the context, answers to division questions may need to be rounded up or rounded down. They explain how they decided whether to round up or down to answer problems eg.: Egg boxes squares of multiples of 10, for example 80 × 80 = 6400. hold 6 eggs. A farmer collects 439 eggs. How many boxes can he fill? OR Egg boxes hold 6 eggs. How many NCETM ACTIVITES boxes must a restaurant buy to have 200 eggs?  solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign  solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates NCETM ACTIVITES Non-statutory guidance Non-statutory guidance Non-statutory guidance Non-statutory guidance Pupils continue to practise their mental recall of multiplication tables when they are calculating Pupils continue to practise recalling and using multiplication tables and related division facts to aid Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Pupils use the whole number system, including saying, reading and writing numbers mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 fluency. Mathematics Appendix 1). They apply all the multiplication tables and related division facts frequently, commit accurately Pupils practise addition, subtraction, multiplication and division for larger multiplication tables. Pupils practise mental methods and extend this to 3-digit numbers to derive facts, (for example 600 ÷ 3 them to memory and use them confidently to make larger calculations. numbers, using the formal written methods of columnar addition and subtraction, short Pupils develop efficient mental methods, for example, using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x = 200 can be derived from 2 x 3 = 6). They use and understand the terms factor, multiple and prime, square and cube numbers. and long multiplication, and short and long division (see Mathematics Appendix 1). 12= 20 x 12 = 240) and multiplication and division facts (e.g. using 3 x 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷3) Pupils practise to become fluent in the formal written method of short multiplication and short division Pupils interpret non-integer answers to division by expressing results in different ways according to the context, They undertake mental calculations with increasingly large numbers and more complex 98 to derive related facts (30 x 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3). with exact answers (see Mathematics Appendix 1). including with remainders, as fractions, as decimals or by rounding (eg, 98 ÷ 4 = = 24 r 2 = 24 ½ = 24.5 ≈ 25). calculations. 4 Pupils develop reliable written methods for multiplication and division, starting with calculations of Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by Pupils continue to use all the multiplication tables to calculate mathematical statements in two-digit numbers by one-digit numbers and progressing to the formal written methods of short 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in order to maintain their fluency. multiplication and division. facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = converting between units such as kilometres and metres. Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, Pupils solve simple problems in contexts, deciding which of the four operations to use and why, 60. Distributivity can be expressed as a(b + c) = ab + ac. 20, 50, etc, but not to a specified number of significant figures. including measuring and scaling contexts, and correspondence problems in which m objects are Pupils solve two-step problems in contexts, choosing the appropriate operation, working with They understand the terms factor, multiple and prime, square and cube numbers and use them to construct Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + connected to n objects (e.g. 3 hats and 4 coats, how many different outfits?; 12 sweets shared increasingly harder numbers. This should include correspondence questions such as the numbers of equivalence statements (eg, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 x 10). 1) x 3 = 9. equally between 4 children; 4 cakes shared equally between 8 children). choices of a meal on a menu, or 3 cakes shared equally between 10 children. Common factors can be related to finding equivalent fractions. Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for

example, 13 + 24 = 12 + 25; 33 = 5 x ). Multiplication structures: to recognise repeated aggregation (addition of groups of the same size); Multiplication structures: to recognise repeated aggregation (addition of groups of the same size); Multiplication structures: to recognise repeated aggregation (addition of groups of the same size); and scaling Multiplication structures: to recognise repeated aggregation (addition of groups of the and scaling (increase/decrease a number of times/by a scale factor of) – both structures should be and scaling (increase/decrease a number of times/by a scale factor of) – both structures should be (increase/decrease a number of times/by a scale factor of) – both structures should be represented as an array. same size); and scaling (increase/decrease a number of times/by a scale factor of) – both represented as an array. represented as an array. Division structures: recognise division as equal sharing; the inverse of repeated aggregation (divided into structures should be represented as an array. Division structures: recognise division as equal sharing; the inverse of repeated aggregation Division structures: recognise division as equal sharing; the inverse of repeated aggregation groups of…) i.e. grouping; and, ratio structure for division i.e. comparison of two quantities – how many times Division structures: recognise division as equal sharing; the inverse of repeated (divided into groups of…) i.e. grouping (divided into groups of…) i.e. grouping less than or more than … aggregation (divided into groups of…) i.e. grouping; and, ratio structure for division i.e. comparison of two quantities – how many times less than or more than … Recording: Following teacher demonstration to link pictorial recording (number line & array), children Recording: Following teacher demonstration to link pictorial recording (number line & array), children Recording: Following teacher demonstration to link pictorial recording (number line & array), children continue to continue to develop recording especially using pictures and their own number lines. They use continue to develop recording especially their own number lines, grid multiplication to represent arrays, develop recording especially grid multiplication to represent arrays, and vertical recording of expanded vertical Recording: Following teacher demonstration to link pictorial recording (number line & standard notation (including the symbols x ÷ and =) and vertical recording of expanded vertical layouts of multiplication & division. They use standard layouts of multiplication & division. They use begin to use standard compact layouts for multiplication & division. array), children continue to develop recording especially grid multiplication to represent VOCABULARY: lots of, groups of; ´, times, multiply, multiplication, multiplied by; multiple of, product, notation (including the symbols x ÷ and =) VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product, once, twice, arrays, and vertical recording of expanded vertical layouts of multiplication & division. repeated addition, array; row, column;, halve; share, share equally; one each, two each, three each… VOCABULARY: lots of, groups of’ times, multiply, multiplication, multiplied by, multiple of, product, three times… ten times…, times as (big, long, wide… and so on), repeated addition, array, row, column, double, They use begin to use standard compact layouts for multiplication & division. group in pairs, threes… tens; equal groups of, divide, division, divided by, divided into, left, left over, once, twice, three times… ten times…times as (big, long, wide… and so on), repeated addition, array, halve, share, share equally one each, two each, three each…, group in pairs, threes… tens equal groups of, VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple remainder row, column, double, halve, share, share equally, one each, two each, three each…, group in pairs, divide, division, divided by, divided into, remainder, factor, quotient, divisible by, inverse of, product, once, twice, three times… ten times…, times as (big, long, wide… and so Language of scaling: once, twice, three times… ten times…; times as (big, long, wide… and so on); threes… tens, equal groups of, divide, division, divided by, divided into, remainder, factor, quotient, Equipment and resources : a range of mathematics equipment as well as everyday situations especially those on), repeated addition, array, row, column, double, halve, share, share equally one each, double, scaling, scale, factor, doubling, trebling, so many times bigger, than (longer than, heavier divisible by two each, three each…, group in pairs, threes… tens, equal groups of, divide, division, that produce arrays e.g. number of seats in an auditorium than, and so on), so many times as much as (or as many as). Inverse ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Multiplication Board (tables divided by, divided into, remainder, factor, quotient, divisible by, inverse Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws, Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws, square); Multiplication grid (grid multiplication) Moving Digits Equipment and resources : a range of mathematics equipment as well as everyday squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes, squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes, http://www.wmnet.org.uk/resources/gordon/Chunking.swf situations especially those that produce arrays e.g. number of seats in an auditorium

ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Grouping; ITPs: Multiplication Array; Multiplication facts; Multiplication tables; Number dials; Multiplication NCETM video examples: Moving from grid to a column method Representing division with place value Multiplication Board (tables square); Multiplication grid (grid multiplication) Moving Multiplication Board (tables square); Multiplication grid (grid multiplication) Board (tables square); Multiplication grid (grid multiplication) Moving Digits counters rapid recall of multiplication facts Digits http://www.wmnet.org.uk/resources/gordon/Chunking.swf http://www.wmnet.org.uk/resources/gordon/Chunking.swf NCETM video examples: Times Tables in Ten Minutes; Grid multiplication as an interim step NCETM video examples: Grid multiplication as an interim step Times Tables in Ten Minutes NCETM video examples: Moving from grid to a column method Representing division with place value counters rapid recall of multiplication facts

Multiplication: example from exemplification (NCETM) Multiplication: example Multiplication: example from exemplification (NCETM) 56 × 27 is approximately 60 × 30 = 1800. Multiplication: example 1328 x 43 1328 x 43 is approximately 1200 × 40 = 48000

Mark drives 19 miles to work every day. He does this on Mondays, Tuesdays, Wednesdays, The class wants to make 235 spiders for a display. How many legs do they need to make? Where possible, model using Base 10 equipment. See Y4 example. Use the equipment to show partitioning Use the equipment e.g. Base 10 materials, to show partitioning i.e. 1328 x 43 = Thursdays and Fridays. How many miles does he travel to work in one week? 235 x 8 is approximately 235 x 10 = 2350 i.e. 56 x 27 = 50 x 20 + 50 x 7 + 6 x 20 + 6 x 7 = (50 + 6) x 27 1000 x 43 + 300 x 43 + 20 x 43 + 8 x 43 = (1000 + 300 + 20 + 8) x 43 Demonstrate using a numberline as ‘repeated’ addition using straws or Base 10 as a model i.e. Model using Base 10 equipment or straws. Use the Leading to formal recording of ‘grid multiplication’ as a representation of the rectangular array. Possible to Leading to recording using ‘grid multiplication’ as a representation of a rectangular equipment to demonstrate partitioning i.e. 235 x 8 = record on a number line – see Y4 example. array. Possible use of Excel to generate array – see Y5 example. 200 x 8 + 30 x 8 + 5 x 8 = (200 + 30 + 5) x 8 Possible use of Excel to generate array to represent 56 x 27 As 1328 x 43 = 43 x x 40 3 56 Record on a number line. 1328 (commutative), 1000 40 000 3000 So, x8 200 30 5 the layout for grid 19 + 19 + 19 + 19 + 19 = partitioned into 10s and 1s as: multiplication can also 300 12 000 900 53 120 10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 = 240 40 be presented 20 800 60 + 3 984 ‘vertically’ as a step 10 9 10 9 10 9 10 9 10 9 0 1600 1840 1880 towards ‘long 8 320 24 57 104 0 multiplication’. 53 120 + 3984 Leading to formal recording of ‘grid multiplication’ 27 as a representation of the rectangular array. 1 3 2 8

x 4 3 x 200 30 5 2 4 3 multiplied by 8 is 24 Leading to an 6 0 3 multiplied by 20 is 60 8 1600 240 40 expanded Then ‘regrouped’ as 10s and 1s: 9 0 0 3 multiplied by 300 is 900 ‘vertical’ layout. 3 0 0 0 3 multiplied by 1000 is 3000 10 + 10 + 10 + 10 + 10 + 9 + 9 + 9 + 9 + 9 = Either begin with 3 2 0 40 multiplied 8 is 320 1600 + 240 + 40 = 1880 so, 235 x 8 = 1880 least or most 8 0 0 40 multiplied by 20 is 800 10 10 10 10 10 9 9 9 9 9 significant figure. 0 1 2 0 0 0 40 multiplied by 300 is 12 000 As 56 x 27 = 27 x 56 x 20 7 4 0 0 0 0 40 multiplied by 1000 is 40 000 5 7 1 0 4 To give a total of 57 104 x 8 (commutative), the layout for 50 1000 350 So, This can be rewritten as: 10 x 5 + 9 x 5 = 50 + 45 = 95 As 235 x 8 = 8 x 235 (commutative), the 2 1 grid multiplication can also be x 5 10 9 200 1600 6 120 42 1120 layout for grid multiplication can also be presented ‘vertically’ (largest 1 3 2 8 45 presented ‘vertically’ as a step towards 30 240 number on the vertical axis) as 1120 + 392 + 392 x 4 3 0 50 95 ‘long multiplication’. 5 40 a step towards ‘long Leading to a compact 1880 1512 ‘vertical’ layout. Either 3 9 8 4 3 multiplied by 1328 is 39 multiplication’. 1 begin with least or most 0 40 multiplied by 1328 is Use informal mental methods of 5 3 1 2 significant figure. 53 120 calculation i.e. PARTITIONING 2 3 5 2 7 19 x 5 = (10 x 5) + (9 x 5) 5 7 1 0 4 To give a total of 1880 x 8 x 5 6 1 1

4 0 8 multiplied by 5 is 40 4 2 6 multiplied by 7 is 42 ITP: Multiplication array Leading to an expanded ‘vertical’ layout:. Leading to an expanded ‘vertical’ Also include examples that involve decimals, money and other measures e.g. 18·6 x 4·4 2 4 0 8 multiplied by 30 is 240 layout. Either begin with least or 1 2 0 6 multiplied by 20 is 120 Leading to formal recording of ‘grid x 19 1 6 0 0 8 multiplied by 200 is 1600 most significant figure. 3 5 0 50 multiplied by 7 is 350 multiplication’ as a representation of 1 8 8 0 To give a total of 1880 1 0 0 0 50 multiplied by 20 is 1000 the rectangular array. (Link with 1 5 1 2 To give a total of 1512

‘area’) 5 10 x 5 = 50 9 x 5 = 45 1

50 + 45 = 95 so, 19 x 5 = 95

2 7

x 5 6 Leading to a compact ‘vertical’ 1 6 2 6 multiplied by 27 is 162 OR COMPENSATION layout: i.e. 19 x 5 = 20 x 5 – 5 1 3 5 0 50 multiplied by 27 is 1350 1 5 1 2 To give a total of 1512 As 18·6 x 4·4 = 4·4 x x 4 0·4 1 18·6 (commutative), the layout for grid 10 40·0 4·00 So, multiplication can also 8 32·0 3·20 74.40 be presented ‘vertically’ as a step towards ‘long 0.6 2·4 0·24 + 7.44 multiplication’. 74·4 + 7·44 81·84

Leading to an expanded ‘vertical’ layout. Either begin with least or most significant figure and then a compact ‘vertical’ layout – see above.

Division: example from exemplification (NCETM) Division example: There are 87 shopping days to Christmas. How many weeks is that? 87  7 Division: example from exemplification (NCETM) Division: example from exemplification (NCETM) Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she have So, ‘87 divided into groups of 7’ is approximately 70 ÷ 7 = 10. 196 ÷ 6 i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30 Each paper clip is made from 9·2 centimetres of wire. What is the greatest number of to buy? paper clips that can be made from 10 metres of wire? GROUPING: Model using bead bar or string. Demonstrate partitioning into groups of 7 using the Leading on from Year 4, divide into ‘groups of the divisor’ using multiplies of groups of 10, 5 and 2 GROUPING i.e. 28 ÷ 6 in this wherever possible and record on a number line. 10m  9·2cm = 1000  9·2 i.e. ‘10m or 1000cm of wire divided into lengths (groups) of interpretation of 87 ÷ 7 as ‘ 87 divided into groups of 7’ 9·2cm’ is approximately 1000 ÷ 10 = 100 ÷ 6 1 1 1 1 context: ‘28 cups divided into groups of 6’.Modelled using ÷6 30 2 r. 4 cups then a bead string or bar 12 r. 4 Leading on from Year 4 and Year 5, divide into ‘groups of the divisor’ using multiplies and recorded on a number of groups of 10, 5 and 2 wherever possible and record on a number line.

0 6 12 18 24 28 line as above. So, 28 ÷ 6 = 4 0 7 14 21 28 35 42 49 56 63 180 771192 196 ÷ 9·2 100 8 4 2 remainder 4 So, ’196 divided into groups of 6 equals 32 remainder 4’ i.e. 196  6 = 32 r.4 or 32 or 32 6 3 73·6 r. 6·4

Record on a number line alongside bead bar or string. 0 7 14 21 28 35 42 49 920 993·6 1000 Also use Grouping ITP: ÷7 1 1 1 1 1 1 1 1 1 1 1 1 When children’s understanding is secure, use vertical recording. Demonstrate vertical recording So, ’1000 divided into lengths of 9·2 equals 108 paper clips with 6.4cm of wire remaining. r. 3 alongside number line recording. Begin to record information vertically introducing ‘repeated 0 7 14 21 28 35 42 49 56 63 70 77 84 87 subtraction/chunking’ (or ‘repeated addition/additive chunking’) lead 3 So, ’87 divided into groups of 7 equals 12 remainder 3’ i.e. 87  7 = 12 r.3 or 12 7 Where necessary 1 9 6 ÷ 6 When children’s understanding is 1 0 0 0  9·2 dividing by ‘groups of secure, use vertical recording. - 9 2 0 100 - 1 8 0 30 the divisor’ using Demonstrate vertical recording 78 10 groups of 10, 5 and 2 1 6 Then begin to divide into ‘groups of the divisor’ using groups of 10, 5 and 2 wherever possible: alongside number line recording. Begin - 4 6 5 (see left) and then to record information vertically 3 1 ÷7 10 2 - 1 2 2 3 4 · 0 EQUAL SHARING e.g. There are 28 paper cups to share equally between 6 people. So if everyone using ‘multiple groups introducing ‘repeated 2 7 ·6 3 has one paper cup, that’s one group of 6 paper cups. Demonstrate recording of one group of 6 paper 14 r. 3 of 10, 5 and 2 (see 4 subtraction/chunking’ (or ‘repeated 6 ·4 cups on a number line….. Another one for each person etc. right) 4 2 addition/additive chunking’). 0 7 14 21 28 35 42 49 56 63 70 77 84 87 so, 196  6 = 32 r.4 or 32 or 32 6 3 So, 6 + 6 + 6 + 6 + 4 = 6 x 4 + 1 = 28 OR so 28 ÷ 6 = 4 r.4 i.e. 28 shared equally between 6 equals 4 so, 1000 9·2 = 108 r.6·4  remainder 4. If children’s understanding is secure, introduce 8 7 ÷ 7

vertical recording e.g. during summer term. - 7 0 10 Demonstrate vertical recording alongside 1 7 number line recording. Begin to record information vertically introducing ‘repeated - 1 4 2 3 subtraction/chunking’ (or ‘repeated 3 addition/additive chunking’. so, 87  7 = 12 r.3 or 12 7

For Year 5 and Year 6: When children are secure with ‘chunking’ and tables facts, introduce short division for single digit divisors using Base 10 equipment or place value counters to secure children’s understanding of the process. For example, 196 ÷ 6 i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30 b. ‘One hundred divided into (groups of) six is difficult using the c. ‘Nineteen tens divided into (groups of) six equals three (groups a. Set out 1 hundred, 9 tens and 6 ones. counters, so exchange 1 hundred for 10 tens. Set out counters of six ‘tens counters’) with one ten remaining’. Set out as indicated and amend written layout. counters as indicated and amend written layout.

d. ‘One ten and six ones divided into (groups of) six is difficult e. ‘Sixteen ones divided into (groups of) six equals two (groups of using the counters, so exchange 1 ten for 10 ones. Set out six ‘ones counters’) with four ones remaining.’ Set out counters as indicated and amend written layout. counters as indicated and amend written layout.

Opportunities for links with other mathematics domains: Opportunities for links with other mathematics domains: Opportunities for links with other mathematics domains: Opportunities for links with other mathematics domains: Cross-curricular and real life connections Making connections to other topics within this year group Making connections to other topics within this year group Making connections to other topics within this year group Learners will encounter aspects of multiplication and division when working on area, relating to Measurement Learners will encounter multiplication and division in: Fractions arrays. Problem solving work involving finding all possibilities and combinations also draws on  Convert between different units of measure [for example, kilometre to metre; hour to Fractions (including decimals and percentages)  identify the value of each digit in numbers given to three decimal places and knowledge of multiplication tables facts. minute] Requirements include: multiply and divide numbers by 10, 100 and 1000 giving answers up to three Fractions work within other curriculum areas and in real life links naturally to multiplication and  solve problems involving converting from hours to minutes; minutes to seconds; years to  multiply proper fractions and mixed numbers by whole numbers, supported by materials and decimal places division work. months; weeks to days. diagrams  multiply one-digit numbers with up to two decimal places by whole numbers The notion of equal groups can emerge in many different activities and contexts, e.g. when packing Fractions  solve problems involving number up to 3 decimal places  use written division methods in cases where the answer has up to two decimal boxes, purchasing quantities of items for several people etc.  recognise and show, using diagrams, families of common equivalent fractions When working on multiplication and division and/or fractions (including decimals and percentages), there are places  count up and down in hundredths; recognise that hundredths arise when dividing an object opportunities to make connections between them, for example:  divide proper fractions by whole numbers [e.g. ⅓ ÷ 2 = ⅙] by one hundred and dividing tenths by ten You could give the children strips of paper and ask them to fold them to show you different proper and mixed  multiply simple pairs of proper fractions, writing the answer in its simplest form 5 3  recognise and write decimal equivalents to 1/4 , 1/2 , 3/4 fractions, for example, ⁄8, 1 ⁄4. Next ask them to multiply these fractions by single digit numbers. They could use When working on multiplication and division and/or fractions there are opportunities to 5  find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the strips to help them: 1 ⁄8 x 6 make connections between them, for example: the digits in the answer as ones, tenths and hundredths Multiply numbers such as: Cross-curricular and real life connections  245.25 by 10, 100 and 1000 Learners will encounter multiplication and division in: 1.35 by 8  Counting – Calculating totals by counting small amounts or a proportion and then scaling up e.g.  ¼ x ½ 5 30 6 5 6 6 3 standing against a tree and using your known height to work out ‘How many of me are equal to the 1x 6 = 6 ⁄8 x 6 = ⁄8 or 3 ⁄8 1 ⁄8 x 6 = 6 + 3 ⁄8 = 9 ⁄8 or 9 ⁄4 Divide numbers such as: height of the tree?’ or counting people on one part of a stadium and multiplying to calculate the total Numbers with decimals are frequently seen in real life, for example when using money, so give the children  12 578 by 10, 100 and 1000 number of spectators. opportunities to multiply these in context. For example, you could give them take-away menus and ask them to  237 by 5 Money – shopping: adding multiple products of the same price, adding coins of same value, working find out how much it would cost to buy four of a meal deal or a particular course. You could give them the total  ⅓ ÷ 2 out fraction/percentage discounts and special offers, sharing bills. cost of six of the same dish and ask to work out which dish you chose. Ratio and proportion Measurement – Scaling quantities (e.g. recipes) to cater for more and less people, reading scales and You could ask the children problems that involve multiplying numbers up to 3 decimal places and link to  solve problems involving the relative sizes of two quantities where missing unlabelled increments on measuring apparatus, calculating area for carpets, decorating etc., scaling measures, such as: values can be found by using integer multiplication and division facts shapes to scale geometric artwork e.g. How would you make this triangle three times its size/half its  Jessie had eight lengths of rope. Each was1m 36cm. If he put them side by side what would the total length When working on multiplication and division and/or ratio and proportion there are size? Comparing river lengths/building heights e.g. the River Nile is x times longer than the River X. be? opportunities to make connections between them, for example: The height of Snowdon is (fraction) of the height of Everest.  Paddy had 12 cartons of orange juice. Each carton contained 0.750l. How much juice did he have Convert the ingredients in this lasagne recipe for 4 people so that it will serve 12: Statistics – Reading scales and determining appropriate scales for different types of graph relating to altogether?  350g minced beef weather, temperature, sound etc., Working with proportion, fractions and percentages using pie charts,  Suzie, the baker, was making 14 loaves of bread for the local supermarket. For each loaf she needed  1 onion comparing data using ratio, fractions and scaling such as proportion of children missing breakfast or 1 1.275kg of flour. What is the total amount of flour that she needed?  1 clove garlic in 7 children under 10 now has a mobile phone etc.  India took part in a sponsored bike ride at her school. She cycled 25 times around the perimeter of the  600g tin of tomatoes

 school playground. The perimeter is 105.34m. How far did she travel?  2 tablespoons tomato puree Measurement  175g lasagne sheets When working on multiplication and division and/or measurement there are opportunities to make connections Algebra between them, for example:  express missing number problems algebraically You could give the children opportunities to rehearse multiplying by 10, 100 and1000 by converting, for example,  use simple formulae millimetres to centimetres, centimetres to metres, metres to kilometres. They could then multiply lengths, masses When working on multiplication and division and/or algebra there are opportunities to make and capacities of different sizes, for example, 14.75kg by 8. You could then put these into problem format, eg: connections between them, for example:  Benji, a party organiser, was going to make a fruit punch. For each guest he needed 0.250ml of orange juice and 0.250l of mango juice. If there are 25 guests coming to the party, what is the total amount of juice Benji Solve missing number problems, e.g. 6(a + 12) = 144 needs? multiply out the equation: 6a + 72 = 144 You could give the children an approximate equivalence between miles and kilometres, for example1.6km is balance by -72: 6a + 72 – 72 = 144 – 72 approximately 1 mile. Then they multiply this amount to find approximate equivalences for other miles, for 6a = 72 example 5 miles, 8 miles, 10 miles, 14 miles. The children could make a spider diagram for this and other Use known division facts: a = 72 ÷ 6 equivalences. a = 12

Find perimeters and areas of rectangles using the appropriate formulae, e.g. a square field has sides of 24.75m. What is its perimeter? What is its area? Measurement  solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate  convert between miles and kilometres When working on multiplication and division and/or measurement there are opportunities to You could give the children lengths of one side of different regular polygons, for example, pentagon, octagon, make connections between them by solving problems such as; decagon, dodecagon and ask them to find their perimeters by multiplying each length by the number of sides the  1 pint = 0.57 litres, how many litres in 8 pints? How many pints in 12 litres? polygon has.  Dan was driving between two cities in France. The sign said the distance was 185km. You could also give the children the lengths of different sized rectangles and ask them to find their areas, for He wanted to know what that was in miles. How can he find out? How many miles is example, a rectangle 28cm by 12cm. it? Set problems involving time and money for the children to use, for example: Statistics  Samir spent 45 minutes completing his homework. It took Pete three times as long. How long did it take Pete  calculate and interpret the mean as an average to complete his homework? When working on multiplication and division and/or statistics there are opportunities to  It took Carol 1 ½ hours to drive from Oxford to London. It took Lorna a third of that time. How long did it take make connections between them, for example: Lorna to travel to London? Solve problems, e.g. find the mean monthly temperature for Reykjavik, Iceland  Harry is given £3.75 a week as pocket money. He is saving it to buy a computer game. How much will he

Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION National Curriculum statements 2014 (statutory) in blue and bold Exemplification and activities taken from NCETM National Curriculum toolkit Year 3 Year 4 Year 5 Year 6 have saved over 8 weeks? What about 12 weeks? Monthly temperatures for Reykjavik  Georgie saved £2.25 of her pocket money each week. How much will she have saved over 9 weeks? Jan Feb March April May June July Aug Sept Oct Nov Dec  Penny had saved £75 over a period of 12 weeks. She saved an equal amount every week. How much did she save each week? -2°C -1°C 3°C 6°C 10°C 13°C 14°C 14°C 11°C 7°C 5°C -2°C Cross-curricular and real life connections Cross-curricular and real life connections Learners will encounter number and place value in: Learners will encounter multiplication and division in: Within the geography curriculum there are opportunities to connect with multiplication and division, for example in Art & Design the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and Within the art and design curriculum there are opportunities to connect with multiplication understanding beyond the local area to include the United Kingdom and Europe, North and South America. This and division, for example in the introduction of the Key Stage 2 Programme of Study it will include the location and characteristics of a range of the world’s most significant human and physical states that pupils should be taught to develop their techniques, including their control and features. Children could, for example, find out about the currencies used in a selection of countries. They could their use of materials, with creativity, experimentation and an increasing awareness of then make up a currency converter using mental calculation strategies and then check using multiplication, for different kinds of art, craft and design. This could include designing and creating life size example: models of, for example a Barbara Hepworth sculpture or a Van Gogh painting where the £1= 1.20 Euros children need to find realistic measurements and then scale them down using division. £2 = 2.40 Euros Geography £3 = 3.60 Euros Within the geography curriculum there are opportunities to connect with multiplication and £4 = 4.80 Euros division, for example in the introduction of the Key Stage 2 Programme of Study it states £5 = 6 Euros that pupils should extend their knowledge and understanding beyond the local area to include the United Kingdom and Europe, North and South America. This will include the location and characteristics of a range of the world’s most significant human and physical features. Work on multiplication and division could include converting between miles and kilometres and vice versa when looking at distances between countries or famous locations, making currency converters for pounds stirling and the currency in the country they are investigating. See, for example:  Mathematics and geography History Within the history curriculum, there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that ‘in planning to ensure the progression described above through teaching the British, local and world history outlined below, teachers should combine overview and depth studies to help pupils understand both the long arc of development and the complexity of specific aspects of the content’. The history curriculum requires that pupils should ‘compare aspects of life in different periods’, suggesting comparisons between Tudor and Victorian periods, for example. Scale models could be one way of learning about life in different periods. See, for example:  The Tudors  The Victorians  The Ancient Egyptians  The Ancient Greeks