Calculation strategies for division

Transcription

1 Calculation strategies for division 1

2 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

3 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

4 Year 2 Group and share using the and = sign Use objects, arrays, diagrams and pictorial representations and grouping on a number line. Mental Calculations: Recall 4 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

5 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

6 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

7 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

8 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 ). They understand what multiplication means, see division as both grouping and sharing, and see division as the inverse of multiplication. Year 4 8

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

10 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 = 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 or to (100 5) (2 5). The array model can help show equivalences. 10

11 Year 5 Divide up to 4 digits by a single digit including those with remainders. Short division with remainders: Now that children are introduced to examples that give rise to remainder answers, division needs to have a real life problem solving context where children consider the meaning of the remainder and how to express it, ie as a fraction, a decimal or as a rounded number or value, depending upon the context of the problem. If children are confident and accurate: Include money and measure contexts. Introduce long division for children who are ready to divide any number by a 2 digit number (eg ) 11

12 Mental Calculations: Count forwards and backwards in steps of powers of 10 for any given number up to 1,000,000. Recall prime numbers to 19. Estimate and use inverse operations to check answers to a calculation. Recall division facts for all numbers up to 12 x 12. Divide numbers mentally drawing upon known facts. Use multiplication and division as inverses. 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, inverse, quotient, prime number, prime factor, composite number Children should: Approximate before they calculate and make this a regular part of their calculating, going back to the approximation to check their answer Key skills for division at Y5: Identify factors, including finding all factor pairs of a number and common factors of 2 numbers. Solve problems where larger numbers are decomposed into their factors. Divide integers and decimals by 10, 100 and Solve problems involving combinations of operations, choosing and using calculations and methods appropriately and using division for scaling by different fractions and problems involving simple rates. Know and use the vocabulary of prime numbers, prime factors and composite numbers. Establish whether a number up to 100 is a prime number. Divide numbers up to 4 digits by a single or 2 digit number using a formal written method including short division and interpret remainders appropriately for the context. Interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, decimals or by rounding. The Big Ideas for Y5: Pupils have a firm understanding of what multiplication and division mean and have a range of strategies for dealing with large numbers, including both mental and standard written methods. They see the idea of factors, multiples and prime numbers as connected and not separate ideas to learn. They recognise how to use their skills of multiplying and dividing in new problem solving situations. Fractions and division are connected ideas: = 36/18 = 2; 13/86 = 1/2 Factors and multiples are connected ideas: 48 is a multiple of 6 and 6 is a factor of

13 Year 6 Divide at least 4 digits by both single and 2 digit numbers (including decimal numbers and quantities) Short division for dividing by a single digit e.g Short division with remainders: Children should continue to use this method, but with numbers to at least 4 digits and understand how to express remainders as fractions, decimals, whole number remainders, or rounded numbers. Real life problem solving contexts need to be the starting point, where children have to consider the most appropriate way to express the remainder. Calculating a decimal remainder: In this example, rather than expressing the remainder as r1, a decimal point is added after the units because there is still a remainder and the one remainder is carried onto zeros after the decimal point (to show there was no decimal value in the original number). Keep dividing to an appropriate degree of accuracy for the problem being solved. Introduce long division by chunking for dividing by 2 digits. Find out How many 36s are in 972? by subtracting chunks of 36 until 0 is reached (or until there is a remainder). Teach children to write a useful list first at the side that will help them decide what chunks to use eg: Useful list: 1 x = 36 10x = x = 3600 Introduce the method in a simple way by limiting the choice of chunks to Can we use 10 lots? Can we use 100 lots? As children become confident with the process, encourage more efficient chunks to get to the answer more quickly (eg 20x, 5x etc) and expand on their useful list. Where remainders occur children should express them as fractions, decimals or use rounding, depending upon the problem. 13

14 Mental Calculations: Recall division facts for all times tables up to 12 x 12 and more complex calculations Perform mental calculations with mixed operations and large numbers. Estimate answers using rounding and approximation and determine levels of accuracy. 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, inverse, quotient, prime number, prime factor, composite number, common factor Children should be able to: Use rounding and place value to make approximations before calculating and use these to check answers against. Key skills for division at Y6: Divide numbers up to 4 digits by a 2 digit whole number using the formal written method of long division and interpret remainders as whole number remainders, fractions or by rounding as appropriate for the context. Use short division where appropriate. Solve multi step problems in a range of contexts choosing appropriate combinations of operations and methods. Round any integer to a required degree of accuracy. Use knowledge of the order of operations to carry out calculations involving the four operations. Identify common factors, common multiples and prime numbers. Use written division methods in cases where the answer has up to 2 decimal places. Solve problems which require answers to be rounded to specified degrees of accuracy. The Big Ideas for Y6: Standard written algorithms use the conceptual structures of the mathematics to produce efficient methods of calculation. There are connections between factors, multiples and prime numbers and between fractions, division and ratios. 14