Lowick and Holy Island C. of E. First Schools

Calculation Policy

INTRODUCTION

This policy has been written in line with the programmes of study taken from the revised National Curriculum for Mathematics (2014). It has been devised to meet the requirements for teaching and learning of mathematics and is designed to ensure pupils have a consistent and smooth progression of learning in calculations across the whole school. The content is set out in staged blocks under the following headings: addition, subtraction, multiplication and division. Statements taken directly from the programmes of study are listed at the beginning of each section.

Age and stage expectations - This policy is organised into age stage expectations as set out in the new National Curriculum 2014, however it is vital that pupils are taught according to the stage that they are currently working at, being moved onto the next level as soon as they are ready, or working at a lower stage until they are secure enough to move on. Our school’s mixed age classes supports such teaching.

Providing a context for calculation - It is important that any type of calculation is based within a real life context or problem solving approach, this enables children to understand the purpose of calculation, and to help them to recognise when to use certain operations and methods when faced with problems. In addition to this, where ever possible, links are made with our thematic/cross curricular approach (Mantle of the Expert). Children need to be taught and encouraged to use the following processes in deciding an approach they will take to a calculation to ensure they select the most appropriate method for the numbers involved:

  • Can I do it in my head using a mental strategy?  
  • Could I use some jottings to help me?
  • Should I use a written method to work this out?

AIMS OF THE POLICY

  • To ensure consistency and progression in our approach to calculation
  • To ensure that children develop an efficient, reliable, formal written method of calculation for all operations
  • To ensure that children can use these methods accurately with confidence and understanding    

 

HOW TO USE THIS POLICY

  • Use the policy as the basis of your planning but ensure you use previous or following years’ guidance to allow for personalised learning
  • Always use Assessment for Learning to identify suitable next steps in calculation for groups of children
  • If, at any time, children are making significant errors, return to the previous stage in calculation
  • Cross reference with mental maths for guidance on key facts, key vocabulary and mental methods
  • Always use suitable resources, models and images to support children’s understanding of calculation and place value, as appropriate
  • Encourage children to make sensible choices about the methods they use when solving problems  

 

EYFS

Early Learning in number and calculation follows the ‘Development Matters’ and towards the ‘Early Years Outcomes’ EYFS documents. This calculation policy is designed to build on progressively from the content and methods established in the Early Years Foundation Stage.

Addition – Early Stages (EYFS)

Children will engage in a wide variety of songs and rhymes, games and activities. They will begin to relate addition to combining two groups of objects, first by counting all and then by counting on from the largest number.  

They will find one more than a given number.

In practical activities and through discussion they will begin to use the vocabulary involved in addition.  

‘You have five apples and I have three apples. How many apples altogether?’    

Subtraction – Early Stages (EYFS)

Children will engage in a variety of counting songs and rhymes and practical activities.   In practical activities and through discussion they will begin to use the vocabulary associated with subtraction.

They will find one less than a given number.

They will begin to relate subtraction to ‘taking away’ using objects to count ‘how many are left’ after some have been taken away.  

6 – 2 = 4     ‘Take two apples away. How many are left?’

Children will begin to count back from a given number.  

 

Multiplication – Early Stages (EYFS)

Children will engage in a wide variety of songs and rhymes, games and activities. In practical activities and through discussion they will begin to solve problems involving doubling.   ‘Three apples for you and three apples for me. How many apples altogether?’

 

Division – Early Stages (EYFS)

Children will engage in a wide variety of songs and rhymes, games and activities. In practical activities and through discussion they will begin to solve problems involving halving and sharing.   Share the apples between two people.   ‘Half of the apples for you and half of the apples for me.’

Calculation Guidelines for Early Years Foundation Stage

ADDITION

SUBTRACTION

MULTIPLICATION

DIVISION

Children begin to record in the context of play or practical activities and problems.

Begin to relate addition to combining two groups of objects

already carried out.

Solve simple word problems using their fingers

Can find one more to ten.

Higher Ability/ Gifted and Talented children progress to using a number line. They jump forwards along the number line using finger.

Begin to relate subtraction to ‘taking away’

activities already carried out

are left.

Can find one less to ten.

Higher Ability/ Gifted and Talented Progression:

Counting backwards along a number line using finger.

Real life contexts and use of practical

equipment to count in repeated groups

Also chanting in 2s, 5s and 10s.

Share objects into equal groups

Use related vocabulary

Activities might include:

  • Sharing of milk at break time
  • Sharing sweets on a child’s birthday
  • Sharing activities in the home corner
  • Count in tens/twos
  • Separate a given number of objects into two groups (addition and subtraction objective in reception being preliminary to multiplication and division)

Count in twos, tens

How many times?

How many are left/left over?

Group

Answer

Right, wrong

What could we try next?

How did you work it out?

Share out

Half, halve

 

 

ADDITION

The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught, and to acquire, secure mental methods of calculation, and one efficient written method of calculation for addition, which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient written method for addition of whole numbers by the end of Year 4. It is however essential, that appropriate mental strategies are taught alongside the written methods in this calculation policy. Note: Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).    

Addition - Stage One

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line. Key skills for addition at Stage 1:

  • Read and write numbers to 100 in numerals, incl. 1—20 in words
  • Recall bonds to 10 and 20, and addition facts within 20
  • Count to and across 100
  • Count in multiples of 1 2, 5 and 10
  • Solve simple 1-step problems involving addition, using objects, number lines and pictorial representations  

Children should be able to - Add with numbers up to 20. Use number lines and number tracks to add by counting on in ones, to start with largest number and count 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. Read and write the addition (+) and equals (=) signs within number sentences. Interpret addition number sentences and solving mixing box problems using concrete objects and number line addition to solve them: 8 + 4 = ___ __ + __   = 6 This should build on prior learning of adding by combining 2 objects.

To support understanding, pupils may physically make and carry calculation with Cuisenaire Rods, Dienes Base material or arrow cards, then compare their practical version to the written form, to help the understanding of it.

Addition - Stage Two

Key vocabulary: add, more, plus, and, make, altogether, total, equal to, equals, double, most, count on, number line, sum, tens, units, partition, addition, column, tens boundary. Key skills for addition at Stage 2:

  • Add a 2-digit number and units (e.g. 27 + 6)
  • Add a 2-digit number and tens (e.g. 23 + 40)
  • Add pairs of 2-digit numbers (e.g. 35 + 47)
  • Add three single-digit numbers (e.g. 5 + 9 + 7)
  • Show that adding can be done in any order (the commutative law)
  • Recall bonds to 20 and bonds of tens to 100 (30 + 70 etc.)
  • Count in steps of 2, 3 and 5 and count in tens from any number
  • Understand the place value of 2-digit numbers (tens and units)
  • Compare and order numbers to 100 using < > and = signs
  • Read and write numbers to at least 100 in numerals and words
  • Solve problems with addition, using concrete objects, pictorial representations, involving numbers, quantities and measures, and applying mental and written methods

Children should be able to - Add with 2-digit numbers: Developing mental fluency with addition and place value involving 2digit numbers, then establish more formal methods.   As with Stage one - to support understanding, pupils may physically make and carry calculation with Cuisenaire Rods, Dienes Base material or arrow cards, then compare their practical version to the written form, to help the understanding of it.

Addition - Stage Three

Key vocabulary: 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, vertical, =carry‘, expanded, compact. Key skills for addition at Stage 3:

  • Read and write numbers to 1000 in numerals and words
  • Add 2-digit numbers mentally, incl. those exceeding 100
  • Add a three-digit number and units mentally (175 + 8)
  • Add a three-digit number and tens mentally (249 + 50)
  • Add a three-digit number and hundreds mentally (381 + 400)
  • Estimate answers to calculations, using 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, units.)
  • Continue to practise a wide range of mental addition strategies, ie. Number bonds, adding the nearest multiple of 10, 100, 100 and adjusting, using near doubles, partitioning and recombining

Children should be able to - Add numbers with up to 3-digits, use the formal written method with the carry going into the next column. Children need to recognise the value of the hundreds, tens and units without recording the partitioning. Pupils need to be able to add in columns. Introduce the expanded column addition method for children who are struggling with understanding of formal written method.  

 

 

Stage One

 

Stage Two

 

Stage Three

+ = signs and missing numbers

Children need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’.

2 = 1+ 1

2 + 3 = 4 + 1

3 = 3

2 + 2 + 2 = 4 + 2

Missing numbers need to be placed in all possible places.

3 + 4 =                     = 3 + 4

3 +  = 7                   7 =  + 4

 + 4 = 7                   7 = 3 + 

 + Ñ = 7                 7 =  + Ñ

The Number Line

Children use a numbered line to count on in ones. Children use number lines and practical resources to support calculation and teachers demonstrate the use of the number line.

7+ 4

+ = signs and missing numbers

Continue using a range of equations as in Stage 1 but with appropriate, larger numbers.

Extend to

14 + 5 = 10 + 

and

32 +  +  = 100   35 = 1 +  + 5

Partition into tens and ones and recombine

12 + 23 = 10 + 2 + 20 + 3

             = 30 + 5

             = 35

Count on in tens and ones

23 + 12 = 23 + 10 + 2

             = 33 + 2

             = 35

The Empty Number Line:

Partitioning and bridging through 10.

The steps in addition often bridge through a multiple of 10

Children should be able to partition the 7 to relate adding the 2 and then the 5.                                               

8 + 7 = 15             

 

 

Add 9 or 11 by adding 10 and adjusting by 1

Add 9 by adding 10 and adjusting by 1

35 + 9 = 44                                   +10

                        

                                                                     -1    

+ = signs and missing numbers

Continue using a range of equations as in Stage 1 and 2 but with appropriate, larger numbers.

Partition into tens and ones

  • Partition both numbers and recombine.
  • Count on by partitioning the second number only e.g.

36 + 53 = 53 + 30 + 6

            = 83 + 6

             = 89

Add a near multiple of 10 to a two-digit number

Secure mental methods by using a number line to model the method. Continue as in Stage 2 but with appropriate numbers

  1. 35 + 19 is the same as 35 + 20 – 1.

Children need to be secure adding multiples of 10 to any two-digit number including those that are not multiples of 10.

48 + 36 = 84

pencil and paper procedures

83 + 42 = 125

either                                                     or

             83                                 80 + 3

       + _42                             + 40 + 2

               5                               120 + 5 = 125

           120

           125

Addition - Stage Four

Key vocabulary: 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, vertical, carry, expanded, compact, thousands, hundreds, digits, inverse. Key skills for addition at Stage 4:

  • Select most appropriate method: mental, jottings or written and explain why
  • Recognise the place value of each digit in a four-digit number
  • Round any number to the nearest 10, 100 or 1000
  • 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, i.e. Number bonds, add the nearest multiple of 10, 100, 1000 and adjust, use near doubles, partitioning and recombining
  • Add numbers with up to 4 digits using the formal written method of column addition
  • 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

Children should be able to - Add numbers with up to 4 digits. Move from expanded addition to the compact column method, adding units first, and ‘carrying’ numbers onto the top line in the correct column. Also include money and measures in contexts.  

Addition - Stage Five

Key vocabulary: 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 at Stage5:

  • Add numbers mentally with increasingly large numbers, using and practising a range of mental strategies i.e. 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 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000
  • Add numbers with more than 4 digits using formal written method of columnar addition.
  • Year 5 Add numbers with more than 4 digits including money, measures and decimals with different numbers of decimal places  

Children should be able to - understand the place value of tenths and hundredths and use this to align numbers with different decimal places. Add numbers with more than 4 digits including money, measures and decimals with different numbers of decimal places.  

Addition - Stage Six

Key vocabulary: 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 at Stage6:

  • 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
  • Pupils understand how to add mentally with larger numbers and calculations of increasing complexity  

Children should be able to - Add several numbers of increasing complexity. 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.
  • Adding several numbers with more than 4 digits.  

 

Stage Four

Stage Five

Stage Six

+ = signs and missing numbers

Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers.

Partition into tens and ones and recombine

Either partition both numbers and recombine or partition the second number only e.g.

55 + 37 = 55 + 30 + 7

             = 85 + 7

             = 92

Add the nearest multiple of 10, then adjust

Continue as in Stage 2 and 3 but with appropriate numbers e.g. 63 + 29 is the same as 63 + 30 - 1

Pencil and paper procedures

367 + 185 = 431

either                     or

   367                         300 + 60 + 7

+185                        100 + 80 + 5  

   12                         400 +140+12 = 552

140

400

552

leading to

   367

+185

   552

     1 1    

Extend to decimals in the context of money.

+ = signs and missing numbers

Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers.

Partition into hundreds, tens and ones and recombine

Either partition both numbers and recombine or partition the second number only e.g.

358 + 73 = 358 + 70 + 3

               = 428 + 3

               = 431

Add or subtract the nearest multiple of 10 or 100, then adjust

Continue as in Stage 2, 3 and 4 but with appropriate numbers e.g. 458 + 79 = is the same as 458 + 80 - 1    

Pencil and paper procedures

Extend to numbers with at least four digits

3587 + 675 = 4262

   3587

+ 675

4262

     1 1 1      

Revert to expanded methods if the children experience any difficulty.

Extend to up to two places of decimals (same number of decimals places) and adding several numbers (with different numbers of digits).

               +54.6

            127.4

             1 1

+ = signs and missing numbers

Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers.

Partition into hundreds, tens, ones and decimal fractions and recombine

Either partition both numbers and recombine or partition the second number only e.g.

                 = 42.8 + 0.3

                 = 43.1

Add the nearest multiple of 10, 100 or 1000, then adjust

Continue as in Stage 2, 3, 4 and 5 but with appropriate numbers including extending to adding 0.9, 1.9, 2.9 etc

Pencil and paper procedures

Extend to numbers with any number of digits and decimals with 1, 2 and/or 3 decimal places.

   13.86

+   9.481

   23.341

     1 1  1

Revert to expanded methods if the children experience any difficulty.

Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level

Extend to decimals with up to 2 decimal

places, including:

  • sums with different numbers of digits;
  • totals of more than two numbers.

Use compensation by adding too much, and then compensating

SUBTRACTION

The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for subtraction which they know they can rely on when mental methods are not appropriate. These notes show the stages in building up to using an efficient method for subtraction of up to 5 whole numbers by the end of Stage 5.

To subtract successfully, children need to be able to: recall all addition and subtraction facts to 20; subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value; partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14). Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction. Using and Applying - Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).                    

Subtraction - Stage One

Key vocabulary: 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_?   Key skills for subtraction at Stage 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, and missing number problems
  • Read and write numbers from 0 to 20 in numerals and words

Children should be able to - Subtract from numbers up to 20. Children consolidate understanding of subtraction practically, showing subtraction on bead strings, using cubes etc. and in familiar contexts, and introduced to more formal recording using number. Subtract by taking away on number lines. Find the distance between two points.   Mental subtraction - Children should start recalling subtraction facts up to and within 10 and 20, and should be able to subtract zero.

Subtraction - Stage Two

Key vocabulary: 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 Stage 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, 100 squares and subtract mentally, including: a two-digit number and units, 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 inverse relationship between addition and subtraction, using this to check calculations and missing number problems
  • Solve simple addition and 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  

Children should be able to - Subtract with 2-digit numbers. Subtract on a number line by counting back, aiming to develop mental subtraction skills. This strategy will be used for: 2-digit numbers subtracting units (by taking away / counting back) e.g. 36—7, 2-digit numbers subtracting tens (by taking away / counting back) e.g. 48—30, subtracting pairs of 2-digit numbers.

Subtraction - Stage Three

Key vocabulary: 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, borrowing, decrease, hundreds, value, digit Key skills for subtraction at Stage 3:

  • Subtract mentally a: 3-digit number and units, 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 21), and select most appropriate methods to subtract, explaining why

Children should be able to - Subtract with 2 and 3-digit numbers. Introduce partitioned column subtraction method.   Approximating before calculating answer should be encouraged. Counting on as a mental strategy for subtraction: Continue to reinforce counting on as a strategy for close-together numbers (e.g. 121—118), and also for numbers that are nearly multiples of 10, 100, 1000 or £s, which make it easier to count on (e.g. 102-89, 131—79, 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:  

 

Stage One

Stage Two

Stage Three

- = signs and missing numbers

7 - 3 =                     = 7 - 3

7 -  = 4                   4 =  - 3

 - 3 = 4                   4 = 7 - 

 - Ñ = 4                   4 =  - Ñ

  • Understand subtraction as 'take away'

    

  • Find a 'difference' by counting up;

I have saved 5p. The socks that I want to buy cost 11p. How much more do I need in order to buy the socks?

  • Use practical and informal written methods to support the subtraction of a one-digit number from a one digit or two-digit number and a multiple of 10 from a two-digit number.

I have 11 toy cars. There are 5 cars too many to fit in the garage. How many cars fit in the garage?

                                                              -5


Use the vocabulary related to addition and subtraction and symbols to describe and record addition and subtraction number sentences

Recording by

- drawing jumps on prepared lines

- constructing own lines

- = signs and missing numbers

Continue using a range of equations as in Stage 1 but with appropriate numbers.

Extend to 14 + 5 = 20 - 

Find a small difference by counting up

42 – 39 = 3

Subtract 9 or 11. Begin to add/subtract 19 or 21

35 – 9 = 26

Use known number facts and place value to subtract(partition second number only)

37 – 12 = 37 – 10 – 2

             = 27 – 2

             = 25

25

27

37


               

32

22

20

15

-10

-2

-5

Bridge through 10 where necessary                             32 - 17                  

- = signs and missing numbers

Continue using a range of equations as in Stage 1 and 2 but with appropriate numbers.

Find a small difference by counting up

Continue as in Stage 2 but with appropriate numbers e.g. 102 – 97 = 5

 

Subtract mentally a ‘near multiple of 10’ to or from a two-digit number

Continue as in Stage 2 but with appropriate numbers e.g. 78 – 49 is the same as 78 – 50 + 1

 

Use known number facts and place value to subtract

Continue as in Year 2 but with appropriate numbers e.g.97 – 15 = 72

     82         87                                 97


             -5

                                         -10

With practice, children will need to record less information and decide whether to count back or forward. It is useful to ask children whether counting up or back is the more efficient for calculations

such as 57 – 12, 86 – 77 or 43 – 28.

 

Pencil and paper procedures

Complementary addition

84 – 56 = 28

                               +20

             +4                                               +4


       56           60                               80           84

Subtraction - Stage Four

Key vocabulary: 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, borrowing, decrease, hundreds, value, digit, inverse. Key skills for subtraction at Stage 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

Children should be able to - Subtract with up to 4-digit numbers Partitioned column subtraction with borrowing. (decomposition): Mental strategies: 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.

Subtraction - Stage Five

Key vocabulary: 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, borrowing, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal. Key skills for subtraction at Stage 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
  • Round any number up to 1 million to the nearest 10, 100, 1000, 10 000 and 100 000

Children should be able to - Subtract with at least 4-digit numbers including money, measures, decimals. Compact column subtraction (with borrowing.). Children who are 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. Subtract with decimal values, including mixtures of integers and decimals, aligning the decimal point.

Subtraction - Stage Six

Key vocabulary: 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, borrowing, decrease, hundreds, value, digit, inverse, tenths, hundredths, decimal point, decimal. Key skills for subtraction at Stage6:

  • 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

Children should be able to - Subtract with increasingly large and more complex numbers and decimal values. Use the compact column method to subtract more complex integers. Use the compact column method to subtract money and measures, including decimals with different numbers of decimal places. 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.

Stage Four

Stage Five

Stage Six

Find a small difference by counting up

This can be modelled on an empty number line (see complementary addition below). Children should be encouraged to use known number facts to reduce the number of steps.

Subtract the nearest multiple of 10, then adjust.

Continue as in Stage 2 and 3 but with appropriate numbers.

Use known number facts and place value to subtract

92 – 25 = 67

Pencil and paper procedures

Complementary addition

754 – 86 = 668

For those children with a secure mental image of the number line they could record the jumps only:
754 – 86 = 668

  

     14 (100)

   600 (700)

      54 (754)

   668

Find a difference by counting up

This can be modelled on an empty number line (see complementary addition below).

Subtract the nearest multiple of 10 or 100, then adjust.

Continue as in Stage 2, 3 and 4 but with appropriate numbers.

Use known number facts and place value to subtract

Pencil and paper procedures

Complementary addition

754 – 286 = 468

OR

754 - 286 = 468                                                                                          

     14 (300)     can be refined to         14 (300)

   400 (700)                                         454 (754)

     54 (754)                                         468

   468

Reduce the number of steps to make the calculation more efficient.

Extend to 2 places of decimals

Find a difference by counting up

  1. 5215

To make this method more efficient, the number of steps should be reduced to a minimum through children knowing:

  • Complements to 1, involving decimals to two decimal places ( 0.16 + 0.84)
  • Complements to 10, 100 and 100

Subtract the nearest multiple of 10, 100 or 1000,

then adjust

Continue as in Stage 2, 3, 4 and 5 but with appropriate numbers.

Use known number facts and place value to subtract

Pencil and paper procedures

Complementary addition

6467 – 2684 = 3783

OR

6467 – 2684 = 3783                                                

       16 (2700)     can be refined to       316 (3000)

     300 (3000)                                     3467 (6467)

   3467 (6467)                                     3783

   3783

Reduce the number of steps to make the calculation more efficient.

Extend to 2 places of decimals

Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level

Mental methods

Use compensation by subtracting too much, and then compensating

Use jottings such as an empty number line to support or explain methods for adding mentally.

Pencil and paper procedures (Written methods)                                                    Subtract more complicated fractions

                                                                                                                                     For Example:


Extend to decimals with up to 2 decimal

places, including:

  • differences with different numbers of

     digits

  • totals of more than two numbers.

Complementary addition

MULTIPLICATION

The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for multiplication which they know they can rely on when mental methods are not appropriate.  

These notes show the stages in building up to using an efficient method for by the end of Stage 4, two-digit by two-digit multiplication by the end of Stage 5, and three-digit by two-digit multiplication by the end of Stage 6.

To multiply successfully, children need to be able to:

  • Recall all multiplication facts to 12 × 12
  • Partition number into multiples of one hundred, ten and one
  • Work out products such as 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using the related fact 7 × 5 and their knowledge of place value
  • Add two or more single-digit numbers mentally
  • Add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value
  • Add combinations of whole numbers using the column method (see above).
  • Use short multiplication to multiply a 1 digit number by a number with up to four digits
  • Use long multiplication to multiply 3 digit and four digit numbers by a number between 11 – 20 by the end of Stage 5
  • Use long multiplication to multiply a two digit number with up to four digits  Use short multiplication to multiply a one digit number by a number with one or two decimal places including money.  

Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication.

Using and Applying - Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).    

Multiplication - Stage One

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count. Key skills for multiplication at Stage 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

Children should be able to - 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 altogether? 3 + 3 + 3 + 3 + 3 = 15  

Give children experience of counting equal group of objects in 2s, 5s and 10s. Present practical problem solving activities involving counting equal sets or groups, as above.

Multiplication - Stage Two

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times as big as, once, twice, three times. Key skills for multiplication at Stage2:

  • 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
  • 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  

Children should be able to - Multiply using arrays and repeated addition (using at least 2s, 5s and 10s). Use repeated addition on a number line: Starting from zero, make equal jumps up on a number line to work out multiplication facts and write multiplication statements using x and = signs.   Use arrays: to help teach children to understand the commutative law of multiplication, and give examples such as 3 x __ = 6. Use practical equipment, use mental recall: - Children should begin to recall multiplication facts for 2, 5 and 10 times tables through practice in counting and understanding of the operation.

Multiplication - Stage Three

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, column, row, commutative, sets of, equal groups, times,         ‘_ times as big as’, once, twice, three times, partition, grid method, multiple, product, tens, units, value Key skills for multiplication at Stage 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 (4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and for missing number problems x 5 = 20, 3 x = 18, x = 32

Children should be able to - Multiply 2-digits by a single digit number. Introduce the grid method for multiplying 2-digit by single-digits. Introduce the grid method with children physically making an array to represent the calculation (e.g. make 8 lots of 23 with 10s and 1s place value counters), then translate this to grid method format. To do this, children must be able to:

  • Partition numbers into tens and units
  • Multiply multiples of ten by a single digit (e.g. 20 x 4) using their knowledge of multiplication facts and place value
  • Recall and work out multiplication facts in the 2, 3, 4, 5, 8 and 10 times tables.
  • Work out multiplication facts not known by repeated addition or other taught mental strategies (e.g. by commutative law, working out near multiples and adjusting, using doubling etc.)

Strategies to support this are repeated addition using a number line, bead bars and arrays.

Stage One

Stage Two

Stage Three

Multiplication is related to doubling and counting groups of the same size.

          

Looking at columns                       Looking at rows

2 + 2 + 2                                         3 + 3

3 groups of 2                                  2 groups of 3

Counting using a variety of practical resources

Counting in 2s e.g. counting socks, shoes, animal’s legs…

Counting in 5s e.g. counting fingers, fingers in gloves, toes…

Counting in 10s e.g. fingers, toes…

Pictures / marks

There are 3 sweets in one bag.

How many sweets are there in 5 bags?

x = signs and missing numbers

7 x 2 =                         = 2 x 7

7 x  = 14                   14 =  x 7

 x 2 = 14                   14 = 2 x 

 x Ñ = 14                   14 =  x Ñ

Arrays and repeated addition

l  l   l   l   4 x 2 or 4 + 4

l  l   l   l

       2 x 4 or 2 + 2 + 2 + 2

Doubling multiples of 5 up to 50

15 x 2 = 30

Partition

Children need to be secure with partitioning numbers into 10s and 1s and partitioning in different ways: 6 = 5 + 1 so

      

       AND double 15

       10     +       5


           

                                     20     +       10      = 30

OR

                                    X     10       5


                                    2     20       10     = 30

x = signs and missing numbers

Continue using a range of equations as in Stage 2 but with appropriate numbers.

Arrays and repeated addition

Continue to understand multiplication as repeated addition and continue to use arrays (as in Stage 2).

Doubling multiples of 5 up to 50

35 x 2 = 70

Partition


            X       30           5


            2       60           10             =70


Use known facts and place value to carry out simple multiplications

Use the same method as above (partitioning), e.g.

32 x 3 = 96                        

= 96

Multiplication - Stage Four

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, array, column, row, commutative, groups of, sets of, lots of, equal groups, times, multiply, times as big as, once, twice, three times... partition, grid method, total, multiple, product, sets of, inverse Key skills for multiplication at Stage 4:

  • Count in multiples of 6, 7, 9, 25 and 1000
  • Recall multiplication facts for all multiplication tables up to 12 x 12
  • Recognise place value of digits in up to 4-digit numbers
  • 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 other strategies mentally 3 x 6 = 6 x 3 , 2 x 6 x 5 = 10 x 6 , 39x7 = 30 x 7 + 9 x 7
  • Solve problems with increasingly complex multiplication in a range of contexts
  • Count in multiples of 6, 7, 9, 25 and 1000
  • Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and units)

Children should be able to - Multiply 2 and 3-digits by a single digit, using all multiplication tables up to 12 x 12 Developing the grid method: Encourage the use of column addition when adding. Move onto short multiplication (see Stage 5) if and when children are confident and accurate multiplying 2 and 3-digit numbers by a single digit this way, and are already confident in carrying for written addition.   To do this, children should: 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 an 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

Multiplication - Stage Five

Key vocabulary: groups of, lots of, times, array, altogether, multiply, count, multiplied by, repeated addition, 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. Key skills for multiplication at Stage 5:

  • Identify multiples and factors, using knowledge of multiplication tables to 12x12
  • 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

Children should be able to - Multiply up to 4-digits by 1 or 2 digits. Be introduced to column multiplication - 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. Use short multiplication for multiplying by a single digit. Pupils could be asked to work out a given calculation using the grid, and then compare it to the column method. What are the similarities and differences? Unpick the steps and show how it reduces the steps. Introduce long multiplication for multiplying by 2 digits. The grid could be used to introduce long multiplication, as the relationship can be seen in the answers in each row. Moving towards more complex numbers.

Multiplication - Stage Six

Key vocabulary: 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 Stage 6:

  • Recall multiplication facts for all times tables up to 12 x 12 (as Y4 and Y5)
  • Multiply multi-digit numbers, up to 4-digit x 2-digit using long multiplication
  • Perform mental calculations with mixed operations and large 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
  • Round any integer to a required degree of accuracy. Suggested Video clips: - Moving from grid method to a compact method (YouTube) - Reinforcing rapid times table recall: (YouTube) - Demonstration of long multiplication (SLEP)  

Children should be able to - use Short and long multiplication as in Stage 5, and multiply decimals with up to 2d.p by a single digit. Use rounding and place value to make approximations before calculating and use these to check answers. Use short multiplication (see Stage 5) to multiply numbers with more than 4-digits by a single digit; to multiply money and measures. Use long multiplication (see Stage 5) to multiply numbers with at least 4 digits by a 2-digit number.

 

Stage Four

Stage Five

Stage Six

x = signs and missing numbers

Continue using a range of equations as in Stage 2 but with appropriate numbers

Partition

Continue to use arrays:

18 x 9 = 162

18 x 9 = (10 x 9) + (8 x 9) = 162

 

OR

Use the grid method of multiplication (as below)

 

Pencil and paper procedures

Grid method

23 x 7 is approximately 20 x 10 = 200

           

            x     20         3


            7   140       21       = 161

Partition

47 x 6 = 282

47 x 6 = (40 x 6) + (7 x 6) = 282

OR

Use the grid method of multiplication (as below)

Pencil and paper procedures

Grid method

72 x 38 is approximately 70 x 40 = 2800

2100 + 60 = 2160

                                           560 + 16 = 576

                                           2160

                                             560 +


                                                     2736

Expanded Column Multiplication

Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by seven’, not ‘three times seven’, although the relationship 3 × 7 should be stressed.

30 + 8                                           38

x   7                                         x 7  

     56   (8 x 7 = 56)                       56

   210   (30 x 7 = 210)                 210

   266                                           266

Partition

87 x 6 = 522

87 x 6 = (80 x 6) + (7 x 6) = 522

OR

Use the grid method of multiplication (as below)

Pencil and paper procedures

Grid method

372 x 24 is approximately 400 x 20 = 8000

Extend to decimals with up to two decimal places.

Short Column Multiplication

The recording is reduced further, with carry digits recorded below the line.

         38

     x   7

      266

         5

Children who are already secure with multiplication for TU × U and TU × TU should have little difficulty in using the same method for HTU × TU or applying decimals.

       286

     x 29

     2574     (9 x 286 = 2574)

     5720     (20 x 286 = 5720)

     8294

     1

Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level

 

Mental methods

Use partitioning

Partition either part of the product e.g. 7.3 x 11   =   (7.3 x 10) + 7.3 = 80.3

OR

Use the grid method of multiplication (as below).

 

 

Pencil and paper procedures (Written methods)

Use written methods to support, record or explain multiplication of:

  • a three-digit number by a two-digit number
  • a decimal with one or two decimal places by a single digit

Grid method

= 49.92

Grid lines can become optional

 
       

DIVISION

The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence. Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation for division which they know they can rely on when mental methods are not appropriate.  

These notes show the stages in building up to long division through Stages 3 to 6 – first long division TU ÷ U, extending to HTU ÷ U, then HTU ÷ TU, and then short division HTU ÷ U. To divide successfully in their heads, children need to be able to:

  • Understand and use the vocabulary of division – for example in 18 ÷ 3 = 6, the 18 is the dividend, the 3 is the divisor and the 6 is the quotient
  • Partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways
  • Recall multiplication and division facts to 10 × 10, recognise multiples of one-digit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value
  • Know how to find a remainder working mentally – for example, find the remainder when 48 is divided by 5
  • Understand and use multiplication and division as inverse operations

Note: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division.

To carry out written methods of division successful, children also need to be able to:

  • Understand division as repeated subtraction
  • Estimate how many times one number divides into another – for example, how many sixes there are in 47, or how many 23s there are in 92
  • Multiply a two-digit number by a single-digit number mentally
  • Subtract numbers using the column method

Using and Applying - Before children move onto the next stage in written calculation it is important that their skills are broadened through their use and application in a range of contexts (including money, time and other measures).

 

 

 

 

 

Division - Stage One

Key Vocabulary: share, share equally, one each, two each…, group, groups of, lots of, array, Key skills for division at Stage 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
  • Children can make connections between arrays, number patterns, and counting in twos, fives and tens

Children should be able to - Group and share large quantities. Using objects, diagrams and pictorial representations to solve problems involving both grouping and sharing. Pupils 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.

Division - Stage Two

Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over Key skills for division at Stage 2:

  • Count in steps of 2, 3, and 5 from 0
  • Recall and use multiplication and division facts for the 2, 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, mental methods, and multiplication and division facts, including problems in contexts

Children should be able to - Group and share, using the ÷ and = sign Use objects, arrays, diagrams and pictorial representations, and grouping on a number line. Pose 12 ÷ 3 as ‘How many groups of 3 are in 12?’

Division - Stage Three

Key Vocabulary: share, share equally, one each, two each…, group, equal groups of, lots of, array, divide, divided by, divided into, division, grouping, number line, left, left over, inverse, short division, carry, remainder, multiple . Key skills for division at Stage 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

Children should be able to - Divide 2-digit numbers by a single digit (where there is no remainder in the final answer). Step 1: When the answer for the first column is zero (1 ÷ 5, as in example), children could initially write a zero above to acknowledge its place, and must always carry the number (1) over to the next digit as a remainder. STEP 2: Pupils move onto dividing numbers with up to 3digits 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 Stage 5.   Include money and measure contexts when confident. 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 .

 

Stage One

Stage Two

Stage Three

Sharing

Requires secure counting skills

-see counting and understanding number strand    

Develops importance of one-to-one correspondence

See appendix for additional information on x and ÷ and aspects of number

               

Sharing – 6 sweets are shared between 2 people. How many do they have each?

        

             lll         lll

Practical activities involving sharing, distributing cards when playing a game, putting objects onto plates, into cups, hoops etc.

Grouping

Sorting objects into 2s / 3s/ 4s etc

How many pairs of socks are there?

There are 12 crocus bulbs. Plant 3 in each pot. How many pots are there?

Jo has 12 Lego wheels. How many cars can she make?

÷ = signs and missing numbers

6 ÷ 2 =                     = 6 ÷ 2

6 ÷  = 3                   3 = 6 ÷ 

 ÷ 2 = 3                   3 =  ÷ 2

 ÷ Ñ = 3                 3 =  ÷ Ñ

Grouping

Link to counting and understanding number strand

Count up to 100 objects by grouping them and counting in tens, fives or twos;…  

Find one half, one quarter and three quarters of shapes and sets of objects

6 ¸ 2 can be modelled as:

There are 6 strawberries.

How many people can have 2 each? How many 2s make 6?

6 ¸ 2 can be modelled as:

                              

              

In the context of money count forwards and backwards using 2p, 5p and 10p coins

Practical grouping e.g. in PE

12 children get into teams of 4 to play a game. How many teams are there?

÷ = signs and missing numbers

Continue using a range of equations as in Stage 2 but with appropriate numbers.

 

Understand division as sharing and grouping

18 ÷ 3 can be modelled as:

Sharing – 18 shared between 3 (see Year 1 diagram)

OR

Grouping - How many 3’s make 18?

   0     3       6       9       12   15     18

    

 

Remainders

16 ÷ 3 = 5 r1

Sharing - 16 shared between 3, how many left over?

Grouping – How many 3’s make 16, how many left over?

   0       3       6       9     12     15 16

Division - Stage Four

Key Vocabulary: share, share equally, one each, two each…, group, equal 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. Key skills needed for division at Stage 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 200 × 3 = 600 so 600 ÷ 3 = 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

Children should be able to - Divide up to 3-digit numbers by a single digit (without remainders initially) Continue to develop short division:   STEP 1: Pupils must be secure with the process of short division for dividing 2-digit numbers by a single digit (those that do not result in a final remainder - see steps in Stage 3), but must understand how to calculate remainders, using this to carry remainders within the calculation process. Short division should only be taught once children have secured the skill of calculating “remainders”. 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.

Division - Stage Five

Key Vocabulary: share, share equally, one each, two each…, group, equal 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 factors, composite number (non-prime). Key skills for division at Stage 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 numbers
  • 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 1000
  • 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. 98 ÷ 4 = 24 r 2 = 24 ½ = 24.5 ˜ 25)
  • Solve problems involving combinations of all four operations, including understanding of the equals sign, and including division for scaling by different fractions and problems involving simple rates.

Children should be able to - Divide up to 4 digits by a single digit, including those with remainders. Short division with remainders: Now that pupils are introduced to examples that give rise to remainder answers, division needs to have a real life problem solving context, where pupils consider the meaning of the remainder and how to express it, i.e. as a fraction, a decimal, or as a rounded number or value, depending upon the context of the problem. The answer to 5309 ÷ 8 could be expressed as 663 and five eighths, 663 r 5, as a decimal, or rounded as appropriate to the problem involved. Include money and measure contexts. See Stage 6 for how to continue the short division to give a decimal answer for children who are confident.   If children are confident and accurate: Introduce long division for pupils who are ready to divide any number by a 2-digit number (e.g. 2678 ÷ 19). This is a Stage 6 expectation.

Division - Stage Six

Key Vocabulary: share, share equally, one each, two each…, group, equal 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 factors, composite number (non-prime), common factor. Key skill for division at Stage 6:

  • 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

Children should be able to - 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.

Stage Four

Stage Five

Stage Six

÷ = signs and missing numbers

Continue using a range of equations as in Stage 2 but with appropriate numbers.

Sharing and grouping

30 ÷ 6 can be modelled as:

grouping – groups of 6 placed on no. line and the number of groups counted e.g.

sharing – sharing among 6, the number given to each person

Remainders

41 ÷ 4 = 10 r1

41 = (10 x 4) + 1

Pencil and paper procedures- Chunking.

72 ÷ 5 lies between 50 ¸ 5 = 10 and 100 ¸ 5 = 20

* Partition the dividend into multiples of the divisor:

                                50 ÷ 5 = 10

                                22 ÷ 5 = 4r2 ® 10 + 4r2 = 14 r 2

                                                OR  

72

-     50     (10 groups)

22

-     20       (4 groups)

                                                               2                          

Answer : 14 remainder 2

Sharing and grouping

Continue to understand division as both sharing and grouping (repeated subtraction).

Remainders

Quotients expressed as fractions or decimal fractions

61 ÷ 4 = 15 ¼ or 15.25

Pencil and paper procedures- Chunking

256 ÷ 7 lies between 210 ¸ 7 = 30 and 280 ¸ 7 = 40

* Partition the dividend into multiples of the divisor:

                                210 ÷ 7 = 30

                                46 ÷ 7 = 6r4 ® 30 + 6r4 = 36r4

OR

                                                      256

   - 210     (30 groups)

                                            46

                                                  -   42       (6 groups)

                                         4                                

Answer: 36 remainder 4

Also, Short Division for More Able Children

               

Considering each column starting from the left. See Stage Six for full explanation.

Sharing, grouping and remainders as Stage Five

Pencil and paper procedures- Chunking

977 ÷ 36 is approximately 1000 ¸ 40 = 25

* Partition the dividend into multiples of the divisor:

                                720 ÷ 36 = 20

                                180 ÷ 36 = 5

                                77 ÷ 36 = 2r5 ® 20 + 5 + 2r5 = 27r5

                                               OR

                                                                       977                                          

-     720     (20 groups)                

                                                               257                                                                                  

                                                                -     180     (5 groups)                    

                                                                     77    

                                                                -       72     (2 groups)

                                                                          5                                      

Answer: 27 5/36

Pencil and Paper procedures- Short Division Method

Write down how many times your divisor goes into the first number of the dividend.If there is a remainder, that's okay.

Write down your remainderto the left of the next digit in the dividend.

  1. Repeat steps 1-3 until you are done.

Both methods above are necessary at this stage, to deal with the wide range of problems experienced at Stage Six.

Calculation Guidelines for Gifted and Talented Children Working Beyond Primary Level

Pencil and paper procedures (Written methods)

Use written methods to support, record or explain division of:

  • a three-digit number by a two-digit number
  • a decimal with one or two decimal places by a single digit.

Refine methods to improve efficiency while maintaining accuracy and understanding.

  1. 6 ÷ 8 is approximately 110 ÷ 10 = 11.

       109.6

   -   80       (10 groups of 8)

       29.6

   -   24         ( 3 )

         5.6

   -     5.6       ( 0.7 )

         0.0

       Answer:   13.7

Pencil and paper procedures (Written methods)

Continue to use the same method as in Year 7 and Year 8. Adjust the dividend and divisor by a common factor before the division so that no further adjustment is needed after the calculation

             e.g.            361.6 ÷ 0.8 is equivalent to 3616 ÷ 8

Use the inverse rule to divide fractions, first converting mixed numbers to improper fractions.

Look at one half of a shape.

How many sixths of the shape can

you see? (six)

So, how many sixths in one half? (three)

So ½ ÷ 1/6 = ½ x 6/1

                                                = 6/2

                                                = 3

Date Policy Adopted by Governing Body

Review dates

April 2015

April 2017

April 2018

Policy developed by: C. Vanson (Headteacher) April 2015

 

 
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