Maths Calculation Policy

Transcription

1 Federation of Trosnant Schools Maths Calculation Policy September 2014

2 This policy contains the key pencil and paper procedures that are to be taught across the Federation of Trosnant Schools. It has been written to ensure consistency and progression throughout the school. Specific strategies have been assigned for calculations to be taught within each year group. However, although these methods will be taught to the majority of children in the year group, it should not be at the expense of genuine understanding and readiness. Children who are confident in using the methods for their year group should be given every opportunity to extend and deepen their understanding through the provision of a rich variety of problem solving activities that allow them to apply their knowledge before being moved on. Although the main focus of this policy is on pencil and paper procedures, it is important to recognise that the ability to calculate mentally lies at the heart of numeracy. Mental calculation is not at the exclusion of written recording and should be seen complementary to, and not as separate, from it. In every written method there is an element of mental processing. Written recording both helps children to clarify their thinking and supports and extends the development of more fluent and sophisticated mental strategies. The long-term aim is for children to be able to select an efficient method, of their choice, that is appropriate for a given task. They should do this by always asking themselves: Can I do it mentally? Can jottings or drawings help me do it mentally? If not, what efficient method can I use? Do I need to use concrete resources to help me with the calculation, or to check the answer I have arrived at?

3 At whatever stage in their learning, and whatever method is being used, children s strategies must still be underpinned by a secure and appropriate knowledge of number facts, along with those mental skills that are needed to carry out the process and judge if it was successful. Daily counting, work on number bonds and multiplication facts is therefore essential in all year groups. Secure understanding of place value should also be constantly reinforced. Wherever possible, calculations should be presented in the form of word problems so children are familiar with choosing correct operations to use depending on the language used in the problem. We will use the RUCSAC acronym across the schools and this should be displayed in all classrooms and consistently reinforced. It is very important that problems are pitched at the right level, using numbers the children are supposed to be working with. It is likely that published materials will have to be adapted for this to be the case. Children should also have regular opportunities to use and apply their number knowledge to extended problem solving activities that are well planned and relevant to their learning journey. Working systematically, showing thoughts and working out through jottings and developing resilience are all key to being successful at this strand of the new curriculum.

4 Year 1 Add with numbers up to 20 Use structured number lines to add, by counting on in ones. Encourage children to look for and start with the larger number in a number sentence, and then count on. Addi on Children should: Have access to a wide range of counting equipment, everyday objects, number tracks and number lines, and be shown numbers in different contexts. Use number tracks/ numberlines to record on (prepared and children s own) Read and write the addition (+) and equals (=) signs within number sentences. Interpret addition number sentences and solve missing box problems using concrete objects and number line addition to solve them: = = = + = 6 This builds on from prior learning of adding by combining two sets of objects into one group (5 cubes and 3 cubes) in Early Years. add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on. Key skills for addition, number and place value at Year 1: Read and write numbers to 100 in numerals, including 1 20 in words Recall number bonds to 10 and 20, and addition facts within 20 Count to and across 100, forwards and backwards, beginning with 1, or from any given number Count in multiples of 1, 2, 5 and 10 Read, write and interpret mathematical statements involving addition (+), subtraction (-) and equals (=) signs Given a number identify one more or one less Solve simple 1-step problems involving addition, using objects, number lines and pictorial representations. Realise the effect of adding zero to a number Know that addition is commutative and can be done in any order (unlike subtraction)

5 Year 2 Add with 2 digit numbers Continue to use practical equipment to support written methods so that children can compare their practical version to the written form. Addi on Add two digit numbers and tens: = Add 2-digit numbers and units: When children are secure working on a structured number line, and with place value to 100, move on to using an unstructured or empty number line. Children should also start jumping in multiples of ten, then multilples of tens and units on their unstructured number line Add pairs of two digit numbers, moving to the partitioned column method when secure adding tens and units: STEP 2: Once children can add a multiple of ten to a 2-digit number mentally (e.g ), they are ready for adding pairs of 2-digit numbers that DO cross the tens boundary (eg ) = STEP 1: Only provide examples that do NOT cross the tens boundary until they are secure with the method itself. N.B. When adding money, children should still use a numberline starting by converting to p sum, tens, units, partition, addition, column, tens boundary Key skills for addition, number and place value at Year 2: Add a 2-digit number and ones (e.g ) Add a 2-digit number and tens (e.g ) Add pairs of 2-digit numbers (e.g ) Add three single-digit numbers (e.g ) Show that adding can be done in any order (the commutative law), and subtraction can not Recall number bonds to 20 and number bonds of tens to 100 (eg ) Count in steps of 2, 3 and 5 and count in tens from any number forwards and backwards Understand the place value of 2-digit numbers (tens and ones) Compare and order numbers to 100 using < > and = signs Round any number to the nearest 10 Read and write numbers to at least 100 in numerals and in words Solve problems with addition, using concrete objects, pictorial representations, involving numbers, quantities and measures, and applying mental and written methods. Recognise and use the inverse relationship between addition and subtraction. use this to check calculations and solve missing number problems

6 Year 3 Add numbers up to 3 digits Introduce the expanded column method: Addi on Carry numbers under bottom Add the units first, in preparation for the compact method In order to carry out this method of addition: Children need to recognise the value of the hundreds, tens and unit without needing to record the partitioning. By the end of Year 3, move to the compact column addition method, with carrying as extension for more able Add units first Children who are very secure and confident with 3-digit expanded column addition should be moved onto the compact column addi on method, being introduced to carrying for the first time. Compare the expanded method to the compact column method to develop an understanding of the process and the reduced number of steps involved.. Remind pupils the actual value is 3 tens add 7 tens (not 3 add 7) which equals 10 tens hundreds boundary, increase, vertical, carry, expanded, compact Key skills for addition, number and place value at Year 3: Read and write numbers to 1000 in numerals and words Add 2-digit numbers mentally, including those exceeding 100 Add a three-digit number and ones mentally ( ) Add a three-digit number and tens mentally ( ) Add a three-digit number and hundreds mentally ( ) Round any number to the nearest 10, 100 Estimate answers to calculations, using the inverse to check answers Solve problems, including missing number problems, using number facts, place value, and more complex addition Recognise place value of each digit in 3-digit numbers (hundreds, tens, ones.) Count from zero in multiples of 4, 8, 50 and 100: find 10 or 100 more or less than a given number Compare and order numbers up to 1000

7 Year 4 Add numbers up to 4 digits Addi on Move from expanded addition to the compact column method, adding units first, and carrying numbers underneath the calculation. eg = 3913 Introduce the compact column addition method by asking children to add two given numbers together using the method they are familiar with (~expanded column addition). Teacher models the compact method with carrying, and asking the children to discuss the similarities and differences and establish how it is carried out. Reinforce correct place value by reminding the children that the Add units actual value is 5 hundreds add 3 first hundreds, not 5 add 3. Carry numbers under bottom line Also include money and measures contexts. For decimal values convert to whole integers and adjust answer as necessary. eg. 1m 34cm + 2m 20cm = 134cm + 220cm = 354cm or 3m 54cm thousands, hundreds, digits, inverse Key skills for addition, number and place value at Year 4: Select most appropriate method: mental, jottings or written and explain why Recognise the place value of each digit in a four-digit number. Count in multiples of 6, 7, 9, 25 and 1000 Count backwards through zero to include negative numbers Round any number to the nearest 10, 100 or Estimate and use inverse operations to check answers Solve 2-step problems in context, deciding which operations and methods to use and why Find 1000 more or less than a given number Continue to practise a wide range of mental addition strategies, ie. number bonds, add the nearest multiple of 10, 100, 1000 and adjust, use near doubles, partitioning and recombining Solve 2-step problems in contexts, deciding which operations and methods to use and why. Estimate and use inverse operations to check answers to a calculation. Read Roman numerals to 100

8 Year 5 Add numbers with more than 4 digits including money, measures and decimals with different numbers of decimal places. Addi on The decimal point should be aligned in the same way as the other place value columns, and must be in the same column in the answer Numbers should exceed 4 digits Pupils should be able to add more that two values, carefully aligning place value columns. Empty decimal places can be filled with zero to show the Say 6 tenths add 7 tenths to reinforce place value decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition, number and place value at Year 5: Add increasingly large numbers mentally, using and practising a range of mental strategies ie. add the nearest multiple of 10, 100, 100 and adjust; use near doubles, inverse, partitioning and re-combining; using number bonds Use rounding to check answers and accuracy Solve multi-step problems in contexts, deciding which operations and methods to use and why Read, write, order and compare numbers to at least 1 million and determine the value of each digit Round any number up to to the nearest 10, 100, 1000, and Interpret negative numbers in context, counting forwards and backwards with positive and negative numbers, including through zero Read Roman numerals to 1000 and recognise years written in Roman numerals

9 Year 6 Add several numbers of increasing complexity Addi on Add several numbers with different numbers of decimal places (including money and measures): Tenths, hundredths and thousandths should be correctly aligned with the decimal point lined up vertically including in the answer row. Zeros could be added into any empty decimal places, to show there is no value to add. Empty decimal places can be filled with zero to show the place value in each column Add several numbers with more than four digits. add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, plus, addition, column, tens boundary, hundreds boundary, increase, carry, expanded, compact, vertical, thousands, hundreds, digits, inverse, decimal places, decimal point, tenths, hundredths, thousandths Key skills for addition, number and place value at Year 6: Perform mental calculations, including with mixed operations and large numbers, using and practising a range of mental strategies Solve multi-step problems in context, deciding which operations and methods to use and why Use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy Read, write, order and compare numbers up to 10 million and determine the value of each digit Round any whole number to a required degree of accuracy Understand how to add mentally with larger numbers and calculations of increasing complexity Use negative numbers in context and calculate intervals across zero

10 Year 1 Subtract from numbers up to 20 Subtrac on Children consolidate understanding of subtraction practically, showing subtraction on bead strings, using cubes etc. and in familiar contexts, and are introduced to more formal recording using number lines. Subtract by taking away Count back in ones on a numbered number line to take away, with numbers up to 20 Find the difference This will be introduced practically with the language find the difference between and how many more? in a range of familiar contexts. Mental subtraction Seven is 3 more than four I am 2 years older than my sister Read, write and interpret number sentences with and = signs 7 4 = 3 Children should start recalling subtraction facts up to and within 10 and 20, and should be able to subtract zero. equal to, take, take away, less, minus, subtract, leaves, distance between, difference, how many more, how many fewer / less than, most, least, count back, how many left, how much less is _? Key skills for subtraction at Year 1: Given a number, say one more or one less. Count to and over 100, forward and back, from any number. Represent and use subtraction facts to 20 and within 20. Subtract with one-digit and two-digit numbers to 20, including zero. Solve one-step problems that involve addition and subtraction, using concrete objects (ie bead string, objects, cubes) and pictures. Solve missing number problems. Read and write numbers from 0 to 100 in numerals and words.

11 Year 2 Subtract with 2 digit numbers Subtrac on Subtract on a number line by counting back, aiming to develop mental subtraction skills. This strategy will be used for: 2-digit numbers subtract units (by taking away / counting back) e.g digit numbers subtract tens (by taking away / counting back) e.g Subtracting pairs of 2-digit numbers on a number line = 24 Move towards more efficient jumps back Then units Subtract 10s first Teaching children to bridge through ten can help them to become more efficient. Mental strategy - subtract numbers close together by counting on eg = 4: Give plenty of opportunity to apply this to money and measures. Children are taught to recognise that when numbers are close together, it is more efficient to count on the difference. They need to be clear about the relationship between addition and subtraction. equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units Key skills for subtraction at Year 2: Recognise the place value of each digit in a two-digit number. Recall and use subtraction facts to 20 fluently, and derive and use related facts up to 100. Subtract using concrete objects, pictorial representations, number lines and mentally, including: a two digit number and ones, a two-digit number and tens, and two two-digit numbers. Show that subtraction of one number from another cannot be done in any order. Recognise and use the inverse relationship between addition and subtraction, using this to check calculations and missing number problems. Solve simple subtraction problems including measures, using concrete objects, pictorial representation, and also applying their increasing knowledge of mental and written methods. Read and write numbers to at least 100 in numerals and in words.

12 Year 3 Subtracting with 2 and 3 digit numbers Introduce partitioned column subtraction method. Subtrac on STEP 1: Introduce this method with examples where no exchanging is required. STEP 2: Introduce exchanging through practical subtraction. Make the larger number with Diennes, then subtract from it. STEP 3: Once pupils are secure with the understanding of exchanging, they can use the partitioned column method to subtract any 2 and 3-digit numbers. Before subtracting 7 from the 72 blocks, they will need to exchange a row of 10 for ten units. Then subtract 7, and subtract 4 tens. Counting on as a mental strategy for subtraction: Continue to reinforce counting on as a strategy for close-together numbers (eg ), and also for numbers that are nearly multiples of 10, 100, 1000 or s, which make it Easier to count on (e.g , , or calculating change from 1 etc.). Start at the smaller number and count on in tens first, then count on in units to find the rest of the difference: equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit Key skills for subtraction at Year 3: Subtract mentally a: 3-digit number and ones, 3-digit number and tens, 3-digit number and hundreds. Estimate answers and use inverse operations to check. Solve problems, including missing number problems. Find 10 or 100 more or less than a given number. Recognise the place value of each digit in a 3-digit number. Counting up differences as a mental strategy when numbers are close together or near multiples of 10 Read and write numbers up to 1000 in numerals and words. Practise mental subtraction strategies, such as subtracting near multiples of 10 and adjusting (e.g. subtracting 19 or

13 Year 4 Subtract with up to 4 digit numbers Subtrac on Give plenty of opportunity to apply this to money and measures. Partitioned column subtraction with exchanging (decomposition) As introduced in Year 3, but moving towards more complex numbers and values. Subtracting money: partition 1.35 into To introduce the compact method, ask children to perform a subtraction with the familiar partitioned column subtraction then display the compact version for the calculation they have done. Ask pupils to consider how it relates to the method they know, what is similar and what is different, to develop an understanding of it. A variety of mental strategies must be taught and practised, including counting on to find the difference where numbers are closer together, or where it is easier to count on. equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is_?, difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse Key skills for subtraction at Year 4: Subtract by counting on where numbers are close together or they are near to multiples of 10, 100 etc. Children select the most appropriate and efficient methods for given subtraction calculations. Estimate and use inverse operations to check answers. Solve addition and subtraction 2-step problems, choosing which operations and methods to use and why. Solve simple measure and money problems involving fractions and decimals to two decimal places. Find 1000 more or less than a given number. Count backwards through zero, including negative numbers. Recognise place value of each digit in a 4-digit number. Round any number to the nearest 10, 100 or 1000 Solve number and practical problems that involve the above, with increasingly large positive numbers.

14 Year 5 Subtract with at least 4 digit numbers Subtrac on Including money, measures and decimals Children who are not still not secure with number facts and place value will need to remain on the partitioned column method until ready for the compact method.subtracting with larger integers Children often get confused when they are subtracting from a number with several zeros eg Use the language of looking at the next neighbour to see if an exchange is possible. In this example, they will need to exchange several times always remembering to reduce the exchanged 10 to 9. The children will need a lot of practise doing this. Subtract with decimal values, including mixtures of integers and decimals aligning the decimal point. Add a zero in any empty decimal places to aid understanding of what to subtract in that column equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is _? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal Key skills for subtraction at Year 5: Subtract numbers mentally with increasingly large numbers. Use rounding and estimation to check answers to calculations and determine, in a range of contexts, levels of accuracy. Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Read, write, order and compare numbers to at least 1 million and determine the value of each digit. Count forwards or backwards in steps of powers of 10 for any given number up to 1 million. Interpret negative numbers in context, counting forwards and backwards with positive and negative integers through 0.

15 Year 6 Subtracting with increasingly large and more complex numbers and decimal values Subtrac on Using the compact column method to subtract more complex integers. Empty decimal places can be filled with zero to show the place value in each column Pupils should be able to apply their knowledge of a range of mental strategies, mental recall skills and informal and formal written methods, when selecting the most appropriate method to work out subtraction problems. equal to, take, take away, less, minus, subtract, leaves, distance between, how many more, how many fewer / less than, most, least, count back, how many left, how much less is _? difference, count on, strategy, partition, tens, units exchange, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal Key skills for subtraction at Year 6: Solve addition and subtraction multi-step problems in context, deciding which operations and methods to use and why. Read, write, order and compare numbers up to 10 million and determine the value of each digit Round any whole number to a required degree of accuracy Use negative numbers in context, and calculate intervals across zero. Children need to utilise and consider a range of mental subtraction strategies, jottings and written methods before choosing how to calculate.

16 Mul plica on Year 1 Multiply with concrete objects, arrays and pictorial representations How many legs will 3 teddies have? There are 3 sweets in one bag. How many sweets are in 5 bags groups of, lots of, times, array, altogether, multiply, count Key skills for multiplication at Year 1: Count in multiples of 2, 5 and 10. Solve one-step problems involving multiplication, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher. Make connections between arrays, number patterns, and counting in twos, fives and tens. Begin to understand doubling using concrete objects and pictorial representations.

17 Mul plica on Year 2 Multiply using arrays and repeated addition (using at least 2, 5 and 10s) 3 x 5 = groups of, lots of, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, double, once, twice, three times... Key skills for multiplication at Year 2: Count in steps of 2, 3 and 5 from zero, and in 10s from any number. Recall and use multiplication facts from the 2, 5 and 10 multiplication tables, including recognising odds and evens. Connect the tables to each other connect the ten times table to place value and the 5 times table to the divisions on a clock. Write and calculate number statements using the x and = signs. Show that multiplication can be done in any order (commutative). Solve a range of problems involving multiplication, using concrete objects, arrays, repeated addition, mental methods, and multiplication facts. Pupils use a variety of language to discuss and describe multiplication.

18 Mul plica on Year 3 partition, grid method, multiple, product, tens, units, value Key skills for multiplication at Year 3: Recall and use multiplication facts for the 2, 3, 4, 5, 8 and 10 multiplication tables, and multiply multiples of 10. Write and calculate number statements using the multiplication tables they know, including 2-digit x single-digit, drawing upon mental methods, and progressing to reliable written methods. Solve multiplication problems, including missing number problems. Develop mental strategies using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240) Solve simple problems in contexts, deciding which operations and methods to use. Develop efficient mental methods to solve a range of problems e.g using commutativity ( = = = 240) and for missing number problems? x 5 = 20, 3 x? = 18,? x = 32 Solve positive integer scaling problems (e.g. 4 times as high), and correspondence problems e.g. how n objects are connected to m objects Through doubling connections between for the 2, 4 and 8 multiplication tables are made

19 Mul plica on Year 4 Multiply 2 and 3-digits by a single digit, using all multiplication tables up to 12 x 12 Children should be able to: - Approximate before they calculate, and make this a regular part of their calculating, going back to the approximation to check the reasonableness of their answer e.g: 346 x 9 is approximately 350 x 10 = 3500 Record the approximation to check the final answer against. - Multiply multiples of ten and one hundred by a single-digit, using their multiplication table knowledge. - Recall all times tables up to 12 x 12 Encourage column addition to add accurately Move on to short multiplication (see Year 5) if and when children are confident and accurate multiplying 2 and 3 digit numbers by a single digit this way, and are confident in carrying for written column Inverse Key skills for multiplication at Year 4: Count in multiples of 6, 7, 9, 25 and 1000 Recall multiplication facts for all multiplication tables up to 12 x 12. Use place value, known facts and derived facts to multiply mentally, e.g. multiply by 1, 10, 100, by 0, or to multiply 3 numbers Use commutativity and factor pairs and other strategies mentally 3 x 6 = 6 x 3, 2 x 6 x 5 = 10 x 6, 39x7 = 30 x x 7 Solve problems with increasingly complex multiplication in a range of contexts

20 Mul plica on Year 5 Multiply up to 4-digits by 1 or 2 digits Moving on to column method... Introduce by comparing a grid method calculation to a short multiplication method, to see how the steps are related, but notice how there are less steps involved in the column method Children need to be taught to approximate first, e.g. for 72 x 38, they will use rounding: 72 x 38 is approximately 70 x 40 = 2800, and use the approximation to check the reasonableness of their answer against. Short multiplication for multiplying by a single digit Pupils could be asked to work out a x x Introduce long multiplication for multiplying 2 digits Moving towards more complex numbers: given calculation using the grid and then compare it to your column what are the similarities and differences? Unpick and show how it 18 x 3 on the 1st row (8 x 3 = 24, carrying the 2 for twenty, then 1 x 3). 18 x 10 on the 2nd row. Put square, factor, integer, decimal, short/long mul plica on, carry Key skills for multiplication at Year 5: Recall multiplication facts for all times tables up to 12 x 12 (as Y4) Identify multiples and factors, including finding all factor pairs of a number Identify common factors of two numbers Know and use the vocabulary of prime numbers, primes number and composite (non-prime) numbers Establish whether a number up to 100 is prime and recall prime numbers up to 19. Solve problems where larger numbers are decomposed into their factors Multiply and divide integers and decimals by 10, 100 and 1000 Recognise and use square and cube numbers and their notation Solve problems involving combinations of operations, choosing and using calculations and methods appropriately

21 Mul plica on Year 6 Short and long multiplication as in Y5, and multiply decimals with up to 2d.p by a single Line up the decimal points in digit the question and answer This works well for money (.p) Remind children and other measures that the single digit belongs in the units column Children will be able to: Use rounding and place value to make approximations before calculating and use these to check answers against. Use short multiplication (see Y5) to multiply numbers with more than 4-digits by a single digit; to multiply money and measures, and to multiply decimals with up to 2d.p. by a single digit. Use long multiplication (see Y5) to multiply numbers with at least 4 digits by a 2-digit number. groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times... partition, grid method, total, multiple, product, inverse, square, factor, integer, decimal, short / long multiplication, carry, tenths, hundredths, decimal Key skills for multiplication at Year 6: Recall multiplication facts for all times tables up to 12 x 12 (as Y4 and Y5) Use knowledge of the order of operations to carry out calculations involving four operations and brackets Identify common factors, common multiples and prime numbers Solve multi-step problems in a range of contexts, choosing appropriate combinations of operations and methods Estimate answers using round and approximation and determine levels of accuracy

22 Year 1 Using objects, diagrams and pictorial representations to solve problems involving both grouping and sharing. Division share, share equally, one each, two each., group, groups of, lots of, array Key skills for division at Year 1 Solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations arrays with the support of the teacher Through grouping and sharing small quantities, pupils begin to understand, division, and finding simple fractions of objects, numbers and quantities. They make connections between arrays, number patterns, and counting in twos, fives and tens.

23 Year 2 Grouping and sharing Division Move on to repeated subtraction on a numberline (no remainders and use multiples the children can easily count in 2s, 3s, 5s, 10s). Subtract groups in equal jumps of the divisor. Pupils could use a bead string or practical apparatus to support. This is an important method to understand division as grouping. Introduce sharing by using Train Tracks method for recording 6 2 = Array, divide, divided by, divided into, division, grouping, number line, left, left over Key skills for division at Year 2: Count in steps of 2, 3, 5, and 10 from 0 Recall and use multiplication and division facts for the 2, 3, 5 and 10 multiplication tables, including recognising odd and even numbers. Calculate mathematical statements for multiplication and division within the multiplication tables and write them using the x, and = signs. Show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot. Solve problems involving multiplication and division, using materials, arrays, repeated addition, repeated subtraction and mental strategies. Methods, and multiplication and division facts, including problems in contexts.

24 Year 3 Divide two and three digit numbers by one or two Introduce division as repeated subtraction on a numberline with remainders. Division Discuss what remainders are: the ones left over that aren t enough to make another group of the divisor. Remainder is written as r Extend to counting back by chunking This method is based on subtracting multiples of the divisor, or chunks. Initially children subtract several chunks, but with practice they should look for the biggest multiples of the divisor that they can find to subtract. Chunking is useful for reminding children of the link between division and repeated subtraction and understanding this is essential for formal long division. Key facts 10 x 6 = 60 Key facts 10 x 6 = 60 Key multiplication facts should be jotted down before children begin chunking 20 inverse, short division, carry, remainder, multiple Key skills for division at Year 3: Recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables (through doubling, connect the 2, 4 and 8s). Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods. Solve problems, in contexts, and including missing number problems, involving multiplication and division. Pupils develop efficient mental methods, for example, using multiplication and division facts (e.g. using 3 2 =6, 6 3 = 2 and 2 = 6 3) to derive related facts (30 2 = 60, so 60 3 = 20 and 20 = 60 3). Pupils develop reliable written methods for division, starting with calculations of 2-digit numbers by 1- digit numbers and progressing to the formal written method of short division.

25 Year 4 Calculate with three digit numbers divided by one digit numbers using short division, and three digit numbers divided by two digit numbers using chunking on a numberline Division 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. Remind children of correct place value, that 96 is equal to 90 and 6, but in short division, pose: - How many 3 s in 9? = 3, and record it above the 9 tens. - How many 3 s in 6? = 2, and record it above the 6 units Once children demonstrate a full understanding of remainders, and also the short division method taught, they can be taught how to use the method when remainders occur within the calculation (e.g ), and be taught to carry the remainder 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 This could be extended to decimals in context of money and measure for the most able N.B. Children should continue to use chunking on a numberline for divisors which have more than one digit divisible by, factor Key skills for division at Year 4: Recall multiplication and division facts for all numbers up to 12 x 12. Use place value, known and derived facts to multiply and divide mentally, including: multiplying and dividing by 10 and 100 and 1. Pupils practise to become fluent in the formal written method of short division with exact answers when dividing by a one-digit number Pupils practise mental methods and extend this to three-digit numbers to derive facts, for example = 600 so = 200 Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as three cakes shared equally between 10 children.

26 Year 5 Calculate with four digit numbers divided by one digit numbers using short division, and three digit numbers divided by two digit numbers using chunking on a numberline When the answer for the first column is zero (1 5, as in example), children could Division The answer to could be expressed as 663 and five eighths, 663 r 5, as a decimal, or rounded as appropriate to the problem involved. initially write a zero above to acknowledge its place, and must always carry the number (1) over to the next digit as a remainder. Short division with remainders: division needs to have a real life problem solving context, where pupils 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. More able children should be moving on to long division (vertical chunking) when they are confidently chunking on a numberline efficiently (using the largest possible multiple of 10 as the initial multiplier) and have rapid recall of all multiplication facts see Year 6 for exemplification of this method. quotient, prime number, prime factors, composite number (non-prime) Key skills for division at Year 5: Recall multiplication and division facts for all numbers up to 12 x 12 (as in Y4). Multiply and divide numbers mentally, drawing upon known facts. Identify multiples and factors, including finding all factor pairs of a number, and common factors of two number. Solve problems involving multiplication and division where larger numbers are decomposed into their factors. Multiply and divide whole numbers and those involving decimals by 10, 100 and Use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers. Work out whether a number up to 100 is prime, and recall prime numbers to 19. Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context Use multiplication and division as inverses. Interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (e.g = 24 r 2 = 241/2 = )

27 Year 6 Divide at least 4 digits by both single-digit and 2-digit numbers (including decimal numbers and quantities) Short division with remainders: Pupils 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 pupils have to consider the most appropriate way to express the remainder. Division Emphasise the need to maintain correct place value so subtraction, and answer, is accurate Calculating a decimal remainder: In this example, rather than expressing the remainder as r 1, 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. Long division relate this directly to chunking on a numberline by subtracting groups of the divisor (model the two methods alongside each other) 20 x 15 8 x 15 Find out How many 15s are in 432? by subtracting chunks of 15, until zero is reached (or until there is a remainder). Remind pupils to write key facts at the side (as they do with numberline division) that will help them decide what chunks to use, e.g.: Expressing with a decimal remainder: 10 x 15 = x 15 = 75 As previously & common factor Key skills for division at Year 6: Here children do how many lots of 15 in 43 write the 2 above and remainder below then bring down the next number to make how many lots of 15 in 132 etc. Recall and use multiplication and division facts for all numbers to 12 x 12 for more complex calculations Divide numbers up to 4 digits by a two-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. Perform mental calculations, including with mixed operations and large numbers. Identify common factors, common multiples and prime numbers. Solve problems involving all 4 operations. Use estimation to check answers to calculations and determine accuracy, in the context of a problem. Use written division methods in cases where the answer has up to two decimal places. Solve problems which require answers to be rounded to specified degrees of accuracy.

28 Year 1 Children to find halves and quarters of whole objects in a range of situations Frac ons Model finding half of regular shapes as dividing the shape in to two equal pieces. ½ ½ Finding a half and a quarter of a quantity Model finding quarter of regular shapes as dividing each of the halves into two equal pieces. ¼ ¼ ¼ ¼ Children to connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole half, halve, quarter, sharing, equal Key skills for factions at Year 1 Recognise, find and name a half as one of two equal part of an object, shape or quantity. Recognise, find and name a quarter as of four equal parts of an object, shape or quantity. Make connections between halves and quarters and the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole.

29 Year 2 Continue to use practical equipment to reinforce ideas of fractions, use correct notation Frac ons Recognise equivalence between 1/2 and 2/4 using fraction cards and lego pieces 1 2 = Move on to finding non-unit fractions of quantities e.g. 2/4 and 3/4 2/4 of 8 = 1/4 + 1/4 3/4 of 8 = 1/4 + 1/4 +1/ = = The shape has four equal parts, the whole has been divided quarters. One of the four parts is shaded so we say 1/4 is shaded, 3/4 is not shaded What fraction of the triangle is blue? What fraction of the circle is not red? fraction, half, halves, quarter, equivalent Key skills for factions at Year 2 Recognise, find, name and write fractions ½, ¼, 2/4, ¾ of a length, shape, set of objects or quantitiy Write simple fractions eg ½ of 6 = 3 and recognise the equivalence of 2/4 and ½ Count in fractions up to 10 starting from any number and using the ½ (1¼, 1 2/4 (or 1½), 1¾, 2

30 Year 3 Continue to use practical equipment, models and images to support understanding. Compare, add and subtract with like denominators Frac ons Recognise both unit and non unit fractions One part out of six parts is shaded so it is 1/6, one out 6 4 out of 9 parts are shaded yellow, which is written 4/9 Add and subtract fractions with the same denominator within 1 whole = 5 5 Use Fractions cards in appendix to aid understanding = Compare and order fractions Use paper and a paperclip as an imaginary numberline. Ask children to show you a half, a quarter, a fifth etc. Use a fraction wall to compare unit and non-unit fractions. Use < and > to compare fractions. Ask pupils to generalise about whether a fractions is greater or less than a half, a quarter or 3/4 tenth, unit fraction, non-unit fraction, denominator, numerator, equivalent Key skills for fractions at Year 3 Count up and down in tenths. Recognise that tenths arise from dividing an object into 10 equal parts and in dividing one digit numbers or quantities in to 10 Recognise, find and write fractions of a discrete set of objects: unit fractions and non unit fractions with small denominators Recognise and use fractions as numbers: unit and non unit fractions with small denominators Add and subtract fractions with the same denominator within one whole (5/7 + 1/7 = 6/7) Compare and order unit fractions, and fractions with the same denominators Solve problems with fractions that involve all of the above

31 including decimals Year 4 Continue to use practical equipment, models and images to support understanding. Compare, add and subtract with like denominators; recognise equivalent fractions Recognise equivalent fractions using a variety of images: Frac ons Add and subtract fractions with the same denominator (can exceed 1 whole): Remember to emphasise the digits moving places to the left or right, NOT the decimal point Solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non unit fractions where the answer is a whole number E.g. A teacher gives out 3/5 of a bag of 40 sweets. How many are left in the bag? Multiply and divide numbers by 10 and 100 using decimal place valuecharts decimal, equivalent, hundredths, denominator, numerator, round, decimal place Key skills for fractions and decimals at Year 4 Recognise and show, using diagrams, families of common equivalent fractions Count up and down in hundredths (recognise that hundredths arise when dividing an object by one hundred Add and subtract fractions with the same denominator Recognise and write decimal equivalents of any numbers of tenths or hundredths Recognise and write decimal equivalents to ¼, ½, ¾ Find the effect of dividing a one or two digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths Round decimals with one decimal place to the nearest whole number Compare numbers with the same number of decimal places up to two decimal places

32 including decimals and percentages Frac ons Year 5 Continue to use practical equipment, models and images to support understanding. Add and subtract fractions; multiply proper fractions and mixed numbers by whole numbers. Recognise mixed numbers and improper fractions. Recognise and convert mixed numbers and improper fractions using lots of real life examples 1 whole one and 3/8 (1 3/8) = 11/8 Multiply proper fractions and mixed numbers by whole numbers Understand that per cent means number of parts per hundred and write as a fraction with a denominator of 100 improper fraction, proper fraction, mixed number, percentage Key skills for fractions, decimals and percentages at Year 5 Compare and order fractions whose denominators are all multiples of the same number Identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths Recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number (2/5 + 4/5 = 6/5 = 1 1/5) Add and subtract fractions with the same denominator and denominators that are multiples of the same number Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams Read and write decimal numbers as fractions (0.71 = 71/100) Recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents Round decimals with two decimal places to the nearest whole number and to one decimal place Read, write, order and compare numbers with up to three decimal places Recognise the per cent symbol (%) and understand that percent relates to number of parts per hundred and write percentages as a fraction with denominator 100, and as a decimal.

33 including decimals and percentages Year 6 Add, subtract, multiply and divide fractions Use arrays to add and subtract fractions with different denominators link to knowledge of equivalent fractions = 31/40 Frac ons Use arrays to multiply fractions Divide proper fractions by whole numbers 1/3 2 = 1/6 Divide into 3 to make thirds then divide every third by 2 Recall and use equivalences between simple fractions, decimals and percentages including different contexts simplest form Key skills for fractions, decimals and percentages at Year 6 Use common factors to simplify fractions Compare and order fractions, including fractions >1 Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions Multiply simple pairs of proper fractions, writing the answer in its simplest form Divide proper fractions by whole numbers (1/3 2 = 1/6) Associate a fraction with division and calculate decimal fraction equivalents (0.375) for a simple fraction (3/8) Identify the value of each digit in numbers given to three decimal places and multiply and divides numbers by 10, 100 and 1000 giving answers up to three decimal places Multiply one digit numbers with up to two decimal places by whole numbers Use written division methods in cases where the answer has up to two decimal places

34 Potential misconceptions with fractions Before teaching fractions it is important to consider where misconceptions may arise and address these from the beginning. Here are some of the most common misconceptions and potential ways to address them. Fractions are read as pieces rather than equal part/ whole relationships Pupils write the shaded part as 3/5. i.e. they see three parts green and five parts white and record these numbers as a fraction. Pupils identify these parts as thirds. Identical fractions of different wholes are not the same Try these activities Ask pupils to discuss whether they would like to share equally with a friend, half of a pouch of 10 50p coins or half of a pouch of 8 1 coins. (The amounts and quantities can be changed according to fluency with calculations). Ask them to explain why. Ask pupils to discuss whether they would prefer to eat half of the cupcake or half of the chocolate cake.

35 The number 1/2 lies half-way between any two labelled integers on a numberline Count from 0 beyond one in different factional steps pointing along an unmarked counting stick. Label 0 and another integer (within the possible range) on the counting stick. Discuss where 1/2 could be and why. Fractions have to be smaller than 1 whole. The larger the denominator the bigger the portion. Provide plenty of opportunity for pupils to share quantities and portions among different sized groups. e.g. if we shared this bag of Jelly Babies with five people how many would each get? If we shared this same bag of jelly babies with six people instead would each person get more or fewer than if there were five people? If we shared this pizza between four people what fraction would each person have? What if it was shared between five people? Would each piece be larger or small than the piece when shared between four people? We have one hour to have turns on the P.E. equipment. Esme is in a group of four will she have more or less goes on the equipment than her friend Paul who is in a group of five?

36 2/3 and 4/6 do not represent the same fraction because the numerators and denominators are different In order for pupils to understand equivalence they must understand that to compare two fractions they must both be fractions of the same whole. One of the first equivalences that pupils should explore is the equivalences of one and other whole numbers. i.e. 1 = 3/3 2 = 4/2 etc. Try this problem from NRICH: Fair feast, which will help pupils experience equivalences of 1/2. Adding the same number to the numerator and dominator will give you an equivalent fraction. e.g. 3/8 = 4/9 Multiplication and division of the numerator and denominator preserve the proportional relationships and produce equivalent fractions but addiing to or subtracting from the numerator and denominator do not preserve the proportional relationship. Provide pupils with opportunities to observe shapes shaded as equivalent fractions. Encourage pupils to compare by asking What do you notice? or What s the same, what s different about these shapes? e.g. Pupils should experience and compare fair sharing situations: Two chocolate bars shared between four children will provide the same number of bars for each person as one chocolate bar shared between two children. Pupils can explore how many chocolate bars are needed for four children or six children etc. Or, one pizza shared equally between three children will give the same number of slices as two pizzas shared between six children. Or, a bag of 40 sweets shared equally between four children will lead to the same number per child as 20 sweets shared equally between two children.

37 When adding (or subtracting) fractions pupils add (or subtract) both the numerators and the denominators. e.g. 5/7 + 1/7 = 6/14 or 5/7 1/7 = 4/0 Pupils do not recognise that the denominator indicates the number of parts of the same whole and therefore treat the two fractions as 4 whole numbers to be added together. Before performing addition and subtraction of fractions, pupils should experience describing part/ whole relationships verbally and in written form, in the same way that they would describe whole number trios. e.g. The yellow and purple shaded parts in the shape below represent 2/5 + 3/5 = 5/5 or 1; or the yellow parts are represented by 1 3/5 or the yellow and purple shaded parts in the shape below represent 2/5 + 2/5 = 4/5 Pupils do not see the need to ensure that the denominators are common before performing addition or subtraction. As above pupils are insecure in the proportional relationship between the numerator and denominator. Pupils need to understand the addition and subtraction of fractions in familiar contexts in order for them to grasp why the denominators must be the same. e.g. Anna and Paul had a bar of white chocolate and a bar of milk chocolate. The bars were the same size. They each ate 14 of the white chocolate and 13 of the milk chocolate. What fraction of the chocolate bars did they each eat? What fraction of the chocolate bars did they both eat altogether? Or Tom and David had a block of butter to make cakes and cookies with. They needed 12 a block for cookies and 13 of a block for the cakes. What fraction of the butter was left after they made their cakes and cookies? Try this challenging problem from NRICH, Andy s Marbles to practice adding and subtracting fractions with different denominators.