Calculation strategies for division 1 Year 1 Group and share small quantities Children will understand equal groups and share items out in play and problems solving. They develop ways of recording calculations using objects, diagrams and pictorial representations to solve problems. 2 Mental Calculations: Count in multiples of 2s, 5s and 10s. Begin to halve numbers using concrete objects and pictorial representations to support. Points to consider: Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array Children should: Use lots of practical apparatus, arrays and picture representations Be taught to understand the difference between ‘grouping’ objects (how many groups of 2 can you make?) and ‘sharing’ (share these sweets between 2 people) Be able to count in multiples of 2s, 5s and 10s. Find half of a group of objects by sharing into 2 equal groups. Key skills for division at Y1: Solve one step problems involving division by calculating the answer using concrete objects, pictorial representations and arrays with support. Through grouping and sharing small quantities, children begin to understand division and finding simple fractions of objects, numbers and quantities. They make connections between arrays, number patterns and counting in 2s, 5s and 10s. The Big Ideas for Y1: The Counting in steps of equal sizes is based on the big idea of ‘unitising’ ; treating a group of, say, five objects as one unit of five. 3 Year 2 Group and share using the ÷ and = sign Use objects, arrays, diagrams and pictorial representations and grouping on a number line. Mental Calculations: • Recall4 and use division facts from the 2, 5 and 10 multiplication tables, including recognising odds and evens. • Show that division can not be done in any order. • Count in steps of 2, 3 and 5 from zero and in 10s from any number, forwards and Points to consider: Key Vocabulary : share, share equally, one each, two each…, group, groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over Children should: Be given practical problem solving activities using concrete objects and pictorial representations wherever possible. Key skills for division at Y2: • Count in steps of 2, 3 and 5 from zero and in 10s from any number, forwards and backwards. • Write and calculate number statements using ÷ and = signs. • Show that multiplication can be done in any order (commutative). • Solve a range of problems involving division, using materials, concrete objects, arrays, repeated addition, mental methods and division facts, including problems in context. • Pupils use a variety of language to discuss and describe division. The Big Ideas for Y2: Pupils should look for and recognise patterns within tables and connections between them (e.g. 5× is half of 10×). Pupils should recognise multiplication and division as inverse operations and use this knowledge to solve problems. They should also recognise division as both grouping and sharing. 5 Year 3 Divide 2 digit numbers by a single digit (where there is no remainder in the final answer) STEP 1: Children continue to work out unknown division facts by grouping on a number line from zero. They are also now taught the concept of remainders, see example. This should be introduced practically and with arrays, as well as being translated to a number line. Children should work towards calculating some basic division facts with remainders mentally for 2s, 3s, 4s, 5s, 8s and 10s, ready for ‘carrying’ remainders across within the short division method. STEP 2: Once children are secure with division as grouping and demonstrate this using number lines, arrays etc short division for larger 2 digit numbers should be introduced, initially with carefully selected examples requiring no calculating of remainders at all. Start by introducing the layout of short division by comparing it to an array. Remind children of correct place value, that 96 is equal to 90 and 6, but in short division pose: * How many 3s in 9? = 3, and record it above the 9 tens * How many 3s in 6? = 2, and record it above the 6 units. 6 STEP 3: Only taught when children can calculate ‘remainders’. Once children demonstrate a full understanding of remainders and also the short division method, they can be taught how to use the method when remainders occur within the calculation (eg 96÷4) and be taught to ‘carry’ the reminder onto the next digit. If needed, children should use the number line to work out individual division facts that occur which they are not yet able to recall mentally. Real life contexts need to be used routinely to help pupils gain a full understanding and the ability to recognise the place of division and how to apply it to problems. 7 Mental Calculations: Recall and use division facts for the 2, 3, 4, 5, 8 and 10 times tables. Develop efficient mental methods to derive related facts (30 x 2 = 60 so 60 ÷3 = 20 and 20 = 60 ÷3). and for missing number problems eg ? x 5 = 20, 3 x ? = 18, ? x ? = 32 Count from 0 in multiples of 4, 8, 50 and 100. Estimate the answer to a calculation and use inverse to check answers. Points to consider: Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, ‘carry’, remainder, multiple Children should be able to: Partition numbers into 10s and units. Key skills for division at Y3: • Write and calculate number statements using the division tables they know, drawing upon mental methods and progressing to reliable written methods. • Solve division problems including missing number problems. • Solve simple problems in contexts deciding which operations and methods to use. • Develop reliable written methods for division, starting with calculations of 2 digit by 1 digit numbers and progressing to the formal written method of short division. The Big Ideas for Y3: It is important for children not just to be able to chant their multiplication tables but also to understand what the facts in them mean, to be able to use these facts to figure out others and to use in problems. It is also important for children to be able to link facts within the tables (e.g. 5× is half of 10×). YearThey understand 4 what multiplication means, see division as both grouping and sharing, and see division as the inverse of multiplication. 8 Year 4 Divide up to 3 digit numbers by a single digit (without remainders initially) Continue to develop short division: STEP 1: Short division should only be taught once children have secured the skills of calculating ‘remainders’. Children must be secure with the process of short division for dividing 2 digit numbers by a single digit but must understand how to calculate remainders using this to ‘carry’ remainders with the calculation process. STEP 2: Children move onto dividing numbers with up to 3 digits by a single digit however problems and calculations provided should not result in a final answer with remainder at this stage. Children who exceed this expectation may progress to Y5 level. When the answer for the first column is zero children could initially write a zero above to acknowledge its place and must always ‘carry’ the number over to the next digit as a remainder. Real life contexts need to be used routinely to help pupils gain a full understanding and the ability to recognise the place of division and how to apply it to problems. Include money and measure contexts. Mental Calculations: Recall division facts for all numbers up to 12 x 12. 9 Count in multiples of 6, 7, 9, 25 and 1000. Use place value, known facts and derived facts to divide mentally eg dividing by 1, 10 and 100. Recognise and use factor pairs in mental calculations. Estimate and use inverse operations to check answers to a calculation. Points to consider: Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, ‘carry’, remainder, multiple, divisible by, factor Children should be able to: Practise to become fluent in the formal written method of short division with exact answers when dividing by a single digit number. Practise mental methods and extend this to 3 digit numbers to derive facts, for example 20 x 3 = 600 so 600 ÷ 3 = 200. Key skills for division at Y4: Solve two step problems in contexts, choosing the appropriate operation, working with increasing harder numbers. Recognise place value of digits in up to 4 digit numbers (thousands, hundreds, tens and units). Solve problems with increasingly complex multiplication in a range of contexts. The Big Ideas for Y4: It is important for children not just to be able to chant their multiplication tables but to understand what the facts in them mean, to be able to use these facts to figure out others and to use them in problems. It is also important for children to be able to link facts within the tables (e.g. 5× is half of 10×). They understand what multiplication means and see division as both grouping and sharing, and to see division as the inverse of multiplication. The distributive law can be used to partition numbers in different ways to create equivalent calculations. For example, 4 × 27 = 4 × (25 + 2) = (4 × 25) + (4 × 2) = 108. Looking for equivalent calculations can make calculating easier. For example, 98 × 5 is equivalent to 98 × 10 ÷ 2 or to (100 × 5) – (2 × 5). The array model can help show equivalences. 10 Year 5 Divide up to 4 digits by a single digit including those with remainders.