Calculation

					Stradbroke Primary School

Calculation Policy

Introduction

Children are introduced to the processes of calculation through practical,
oral and mental activities. As children begin to understand the underlying
ideas they develop ways of recording to support their thinking and
calculation methods, use particular methods that apply to special cases,
and learn to interpret and use the signs and symbols involved. Over time
children learn how to use models and images, such as empty number lines,
to support their mental and informal written methods of calculation. As
children's mental methods are strengthened and refined, so too are their
informal written methods. These methods become more efficient and
succinct and lead to efficient written methods that can be used more
generally. By the end of Year 6 children are equipped with mental, written
and calculator methods that they understand and can use correctly. When
faced with a calculation, children are able to decide which method is most
appropriate and have strategies to check its accuracy. At whatever stage
in their learning, and whatever method is being used, it 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.

The overall aim is that when children leave primary school they:

      have a secure knowledge of number facts and a good understanding
     of the four operations;
      are able to use this knowledge and understanding to carry out
     calculations mentally and to apply general strategies when using one-
     digit and two-digit numbers and particular strategies to special cases
     involving bigger numbers;
      make use of diagrams and informal notes to help record steps and
     part answers when using mental methods that generate more
     information than can be kept in their heads;
      have an efficient, reliable, compact written method of calculation
     for each operation that children can apply with confidence when
     undertaking calculations that they cannot carry out mentally;
Mental strategies need to be established. These will be based on a solid
understanding of place value and a sense of number. It is important that
children's mental methods of calculation are practised and secured
alongside their learning and use of efficient written methods for
calculation.

Mental strategies involve:

       Remembering number facts and recalling them without hesitation
       Doubling and halving
       Using known facts to calculate unknown facts
       Re-ordering numbers
       Understanding and using relationships between addition & subtraction,
        multiplication and division
       Checking answers
       Estimating answers
       Seeing/breaking numbers up in different ways eg 64 = 50 + 14

Children will need a repertoire of mental strategies in order to solve
calculations in their heads:
 Count on /back in 1s, 10s, 100s from any start point
 doubles / near doubles
 bridging 10 / bridging 20
 adding 9 by +10 & -1 (compensation)

Stages in Addition

To add successfully, children need to be able to:

        recall all addition pairs to 9 + 9 and complements in 10;
        add mentally a series of one-digit numbers, such as 5 + 8 + 4;
        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;
        partition two-digit and three-digit numbers into multiples of 100, 10
         and 1 in different ways
Informal jottings will help children with their calculations.

Stage one




They use numberlines and practical resources to support calculation and
teachers demonstrate the use of the numberline.

3+2=5
                         +1       +1

___________________________________________
0   1    2   3   4   5   6   7  8    9

Children then begin to use numbered lines to support their own
calculations using a numbered line to count on in ones.

8 + 5 = 13
                                       +1 +1 +1 +1   +1


0    1   2   3   4   5   6    7   8     9   10 11 12 13 14 15

Children will begin to use ‘empty number lines’ themselves starting with
the larger number and counting on.

Stage two The empty number line

    The mental methods that lead to column addition generally involve
     partitioning, e.g. adding the tens and ones separately, often starting
     with the tens. Children need to be able to partition numbers in ways
     other than into tens and ones to help them make multiples of ten by
     adding in steps.
    The empty number line helps to record the steps on the way to
     calculating the total.
Steps in addition can be recorded on a number line. The steps often
bridge through a multiple of 10.

8 + 7 = 15



48 + 36 = 84



or:




WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS?

Addition and subtraction

Do they know addition and subtraction facts to 20?
Do they understand place value and can they partition numbers?
Can they add three single digit numbers mentally?
Can they add and subtract multiples of 10?
Can they add and subtract any pair of two digit numbers mentally?
Can they explain their mental strategies orally and record them using
informal jottings?

Stage 3: Partitioning

       The next stage is to record mental methods using partitioning. Add
      the tens and then the ones to form partial sums and then add these
      partial sums.

     Partitioning both numbers into tens and ones mirrors the column
      method where ones are placed under ones and tens under tens. This
      also links to mental methods.

Record steps in addition using partitioning:

47 + 76 =

40 + 70 =
7+6=

47 + 76 = 47 + 70 + 6 = 117 + 6 = 123
47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123

47 + 76

Stage 4: Expanded method in columns

      Move on to a layout showing the addition of the tens to the tens
     and the ones to the ones separately. To find the partial sums either
     the tens or the ones can be added first, and the total of the partial
     sums can be found by adding them in any order. As children gain
     confidence, ask them to start by adding the ones digits first always.
      The addition of the tens in the calculation 47 + 76 is described in
     the words 'forty plus seventy equals one hundred and ten', stressing
     the link to the related fact 'four plus seven equals eleven'.

The expanded method leads children to the more compact method so that
they understand its structure and efficiency. The amount of time that
should be spent teaching and practising the expanded method will depend
on how secure the children are in their recall of number facts and in their
understanding of place value.

Write the numbers in columns.

Adding the tens first:




Adding the ones first:




Discuss how adding the ones first gives the same answer as adding the
tens first. Refine over time to adding the ones digits first consistently.


Stage 4: Column method
      In this method, recording is reduced further. Carry digits are
     recorded below the line, using the words 'carry ten' or 'carry one
     hundred', not 'carry one'.

Later, extend to adding three two-digit numbers, two three-digit
numbers and numbers with different numbers of digits.




Stage 4




Column addition remains efficient when used with larger whole numbers
and decimals. Once learned, the method is quick and reliable.




Stages in Subtraction

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, f16 - 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).
Stage 1
Children are encouraged to develop a mental picture of the number
system in their heads to use for calculation. They develop ways of
recording calculations using pictures etc.




Informal jottings will help children with their calculations
They use numberlines and practical resources to support calculation.
Teachers demonstrate the use of the numberline.

Stage 2
6–3=3
              -1 -1 -1
___________________________________
0   1  2 3 4 5 6 7 8 9 10

The numberline should also be used to show that 6 - 3 means the
‘difference between
6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are
apart.



0    1    2       3       4       5   6   7   8       9    10

Children then begin to use numbered lines to support their own
calculations - using a numbered line to count back in ones.

13 – 5 = 8

                                                      -1    -1 -1   -1 -1
.
     0    1   2       3       4   5   6   7   8   9       10 11 12 13 14 15

Counting back
    First counting back in tens and ones.

47 – 23 = 24


                -1        -1        -1    - 10           - 10


          24        25     26        27         37              47


    Then helping children to become more efficient by subtracting the
     units in one jump (by using the known fact 7 – 3 = 4).

47 – 23 = 24

                                         -10         -10
                               -3

                          24        27         37          47


    Subtracting the tens in one jump and the units in one jump.

47 – 23 = 24

                                                -20
                                -3


                           24 27                           47




    Bridging through ten can help children become more efficient.


42 – 25 = 17

                                          -20
                     -3        -2



               17     20        22                  42
Counting on

If the numbers involved in the calculation are close together or near to
multiples of 10, 100 etc, it can be more efficient to count on.

Count up from 47 to 82 in jumps of 10 and jumps of 1.



 82 - 47

                                   +10     +10     +10   +1 +1
                        +1 +1 +1


                    47 48 49 50       60     70      80 81 82


 Help children to become more efficient with counting on by:

    Subtracting the units in one jump;
    Subtracting the tens in one jump and the units in one jump;
    Bridging through ten.




The children must begin to learn subtraction through their use of number
lines. They will learn to calculate the difference between numbers. This
method is essential and can be used with all ages.
WHEN ARE CHILDREN READY FOR WRITTEN CALCULATIONS?

Addition and subtraction

Do they know addition and subtraction facts to 20?
Do they understand place value and can they partition numbers?
Can they add three single digit numbers mentally?
Can they add and subtract multiples of 10?
Can they add and subtract any pair of two digit numbers mentally?
Can they explain their mental strategies orally and record them using
informal jottings?

Stage 3
DECOMPOSITION – EXPANDED METHOD

This method makes the move to traditional decomposition easier. The
children must understand the value of the digits they are using and
develop an appreciation of the need to take from the next column and the
effect this has on the sum. At this stage the children must be taught to
set out their work accurately.




Stage 4
TRADITIONAL DECOMPOSITION

This is only taught once the children have mastered the above methods.
By this stage the children should be able choose the most efficient
method for the calculation they have to complete.

                4       15       1
                    5        6   3

               - 2           7   8

                    2        8   5
                          Stages in Multiplication

To multiply successfully, children need to be able to:

      recall all multiplication facts to 10 × 10;
      partition number into multiples of one hundred, ten and one;
      work out products such as 70 × 5, 70 × 50, 700 × 5 or700 × 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).
      Multiplication is linked to division
      Count numbers in groups
      Recognise that multiplication can be done in any order –
     commutative law

Stage 1
Children will experience equal groups of objects and will count in 2s and
10s and begin to count in 5s. They will work on practical problem solving
activities involving equal sets or groups.




Stage 2
Children will develop their understanding of multiplication and use
jottings to support calculation:

    Repeated addition

3 times 5   is   5 + 5 + 5 = 15   or 3 lots of 5 or 5 x 3
Repeated addition can be shown easily on a number line:

5x3=5+5+5

                    5                           5                  5



    0   1       2       3   4       5   6   7       8   9   10 11 12 13 14 15


    5 x3=5+5+5

    Commutativity

Children should know that 3 x 5 has the same answer as 5 x 3. This can
also be shown on the number line.


                    5                           5                  5



    0   1       2       3   4       5   6   7       8   9   10 11 12 13 14 15

            3                   3               3           3          3


       Arrays

Children should be able to model a multiplication calculation using an
array. This knowledge will support with the development of the grid
method.




                                            5 x 3 = 15


                    3 x 5 = 15
Before moving pupils to stage 3:

Do they know the 2, 3, 4, 5 and 10 times table
Do they know the result of multiplying by 0 and 1?
Do they understand 0 as a placeholder?
Can they multiply two and three digit numbers by 10 and 100?
Can they double and halve two digit numbers mentally?
Can they use multiplication facts they know to derive mentally other
multiplication facts that they do not know?
Can they explain their mental strategies orally and record them using
informal jottings?

The above lists are not exhaustive but are a guide for the teacher to
judge when a child is ready to move from informal to formal methods of
calculation.

Stage 3
The children should be taught the basic mental method using partitioning.
It is essential that at this stage they are taught to estimate their
answer.

For example

Estimate: 30 x 5 = 150   Actual: 27 x 5= (20 x 5) + (7 x 5)
                                         100 + 35 = 135



Stage 4
Once the children have grasped this method they should be taught the
grid method. The children should practice 2digit x 1 digit (no carrying)
before moving forward to gradually introduce carrying and finally moving
to 3 digit x 1 digit.

Example: 23 x 8
Estimate: 20 x 8 = 160

Actual:        x 20 3
               8 160 24 184
The children should be taught how to work with larger numbers. The
principles of estimate and check (inverse operation) should also be
modelled.

Example: 157 x 23
Estimate: 160 x 20 = 320
Actual:
 x 100      50     7
 20 2000 1000 140 3140
 3 300 150 21            471
                        3611

The children should be taught how to work with larger numbers and
decimals. The principles of estimate and check (inverse operation) should
also be modelled.

Example: 27.5 x 12 = 330
Estimate: 30 x 12 = 360
Actual:
 x   20     7    0.5
 10 200 70        5    275
 2   40    14     1     55
                       330

Stage 5
Once the children are confident users of the grid method they should be
taught how to use the traditional vertical format. The children should be
shown the two methods alongside each other to enable the children to
identify the similarities between the formats. Again the children should
be taught the expanded methods before moving to the compact method in
step six.

Example: 124 x 6 =
Estimate: 120 x 6 = 720 Actual: 124
                               X6
                                24 (6 x 4)
                              120 (6 x 20)
                              600 (6 x 100)

                                744
Stage 6
This final step includes the use of both compact and traditional ‘long
multiplication’. It is essential that the children recognise the similarities
with the grid method and so we should continue to make teaching
comparisons.

Example: 124 x 6 =
Estimate: 120 x 6 = 720 Actual: 124
                               X6

                                  744

                                  12



Example: 124 x 25 =
Estimate: 120 x 30 = 3600
Actual:                          124
                                 X25

                                  620    (5 x 124)
                                 2480    (20 x 124)
                                 3100
                                  11




Stages in Division

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.
      Dividing by 1 leaves the number unchanged


Stage 1
Children will understand equal groups and share items out in play and
problem solving. They will count in 2s and 10s and later in 5s.




Children will develop their understanding of division and use jottings to
support calculation

    Sharing equally

6 sweets shared between 2 people, how many do they each get?




    Grouping or repeated subtraction

There are 6 sweets, how many people can have 2 sweets each?
Stage 2

Repeated subtraction using a number line or bead bar

12 ÷ 3 = 4




  0      1   2   3   4    5        6    7    8    9        10 11 12


         3                 3                       3                  3




The bead bar will help children with interpreting division calculations such
as 10 ÷ 5 as ‘how many 5s make 10?’

This builds upon a mental method. The children should be taught the
repeated subtraction method using a number line.



Stage 3
The children should understand the links with multiplication and would be
able to use their table knowledge and numbers lines to make larger jumps.
When moving onto the vertical method the children must be given the
opportunity to begin with sums they can do mentally. They should also be
taught to estimate and check using the inverse operation.

Example: 62 / 6 = 10 r 2
Estimate: 60 / 6 = 10           Actual: 6 / 62
                                   - 60      (10 x 6)
                                      2
For larger numbers, when it becomes inefficient to count in single
multiples, bigger jumps can be recorded using known facts.

Example without remainder:
81 ÷ 3               30                30         21

                          10            10             7
                 0            30             60            81
This could either be done by working out the numbers of threes in each
jump as you go along (10 threes are 30, another 10 threes makes 60, and
another 7 threes makes 81. That’s 27 threes altogether) or by counting in
jumps of known multiples of 3 to reach 81 (30 + 30 + 21) then working out
the number of threes in each jump.

Example with remainder:
158 ÷ 7           70         70         14     2

                  10        10          2
              0        70         140        154 158

10 sevens are 70, add another 10 sevens is 140, add 2 more sevens is 154
add 2 makes 158. So there are 22 sevens with a remainder of 2.
The remainder is indicated above the jump rather than inside it, so that
children do not mistakenly add 10, 10, 2 and 2 and get an answer of 24.




The same method should be used with larger numbers.

Example: 162 / 6 = 27
Estimate: 160 / 10 = 16     Actual: 6 / 162
                                      - 60      (10 x 6)
                                       102
                                      - 60 (10 x 6)
                                        42
                                       -42 (7 x 6)
                                         0

Stage 4

Once pupils are more confident with mental strategies they should be
taught the contracted method.



Example: 163 / 6 = 27 r1
Estimate: 160 / 10 = 16 Actual: 6 / 163
                                   -120 (20 x 6)
                                    43
                                 - 42 (7 x 6)
                                      1
Stage 5
This method should be extended further to include larger numbers and
decimals.
Children will continue to use written methods to solve short division TU ÷
U.

Children can start to subtract larger multiples of the divisor, e.g. 30x

Short division HTU ÷ U

196 ÷ 6
                  32 r 4
             6 ) 196
               - 180     30x6
                  16
               -  12     2x6
                  4

             Answer :      32 remainder 4 or       32 r 4




Children will continue to use written methods to solve short division TU ÷
U and HTU ÷ U.

Long division HTU ÷ TU

972 ÷ 36
                  27
           36 ) 972
               - 720       20x36
                 252
               - 252       7x36
                   0

             Answer :      27

Any remainders should be shown as fractions, i.e. if the children were
dividing 32 by 10, the answer should be shown as 3 2/10 which could then
be written as 3 1/5 in it’s lowest terms.
                                  Summary

       children should always estimate first

       always check the answer, preferably using a different method eg.
        the inverse operation

       always decide first whether a mental method is appropriate

       pay attention to language - refer to the actual value of digits

       children who make persistent mistakes should return to the
        method that they can use accurately until ready to move on

       children need to know number and multiplication facts by heart

       discuss errors and diagnose problem and then work through
        problem - do not simply re-teach the method

       when revising or extending to harder numbers, refer back to
        expanded methods. This helps reinforce understanding and
        reminds children that they have an alternative to fall back on if
        they are having difficulties.



Can I   do it in my head?
Can I   change it to make it easier?
Can I   use jottings to help me work it out?
Can I   use a calculator?

				
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