WithernseaPrimary School

Calculations 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. They will do this by asking themselves:

  • Can I do this in my head?
  • Can I do this in my head using drawing or jottings?
  • Do I need to use a pencil and paper procedure?
  • Do I need a calculator?

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 and reliable written method of calculation for each operation that children can apply with confidence when undertaking calculations that they cannot carry out mentally;
  • use a calculator effectively, using their mental skills to monitor the process, check the steps involved and decide if the numbers displayed make sense.

Written methods for addition of whole numbers

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 supported be informal calculation strategies. These notes show the stages in building up to using an efficient written method for addition of whole numbers.

 

To add successfully, children need to be able to:

  • recall all addition pairs to 9 + 9 and complements in 10 e.g. 6 + 4 = 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.

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 addition.

Resources – ITP’s, number line, place value cards, hundred square, numicom, ITPs.

 

Stage 1

 

    

At   first children will relate addition to the combining of 2 groups:

 

Count   out 3, count out 2.

Put   together and count out 5

 

Children are encouraged   to make use of fingers as these are a constantly available resource for   calculations at this level.

 

For example: 3 + 2 = 5

 

                                                                                +           = 5

 

Alternatively, count out 3 and then count on 2 more to make 5

 

Stage 2

 

 

The next step is to be able to count one   more, and then several more, on a number line:

 

 

 

 

 

 

 

 

 

 

 

For example:  6 + 1 = 7

 

                                                                                     +1

                                  

                      
      
      

 

      
      

                           

                      
      
      

0     1     2      3      4   

      
      

                     

                      
      
      

         5      6      7      8

      
      

                     

                      
      
      

 

      
      

                                                                                                                                                                                                                                                                                                                      

 

 

Stage 3

 

Steps   in addition can be recorded on a number line. The steps often bridge through   a multiple of 10.

 

The mental   methods involve partitioning, e.g. adding the tens and ones separately,   often starting with the tens.

The empty number line   helps to record the steps on the way to calculating the total.

(      Year 2)

 

 

 

This method can be used for the   addition of larger numbers and decimals further up the school.

8 + 7 = 15

        

48 + 36 = 84

        

or:
      

 

 

 

        

                  
     
     

+7

     
     

            

                  
     
     

+0.3

     
     

    35.8 + 7.3 = 43.1

 

                                  

                      
      
      

 

      
      

                           

                      
      
      

35.8

      
      

                     

                      
      
      

 

      
      

                        

                      
      
      

42.8

      
      

                     

                      
      
      

 

      
      

                        

                      
      
      

43.1

      
      

                     

                      
      
      

 

      
      

                                                                 

                      
      
      

 

      
      

                        

                      
      
      

 

      
      

            

 

Stage 4

 

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.

 

 

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.

 

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’

(      Year 4)

 

Record   steps in addition using partitioning:

   

47 + 76 = 40 + 70 =    110 

                    7 + 6 =    13 

                                  123

 

Three digit numbers:      

187          or          187

246                       274

300                       300

120                       150

  13                         11

400                       461

30

3

433

 

Decimals

14.28

17.56

20.00

11.00

  0.70

  0.14

31.84

 

Stage 5

 

Carrying

Children should not be taught this very refined column   method until they are a secure level 4 and not before Year 5. They can be taught this if they haven’t   reached this level after SATs to support their transfer to High School.

 

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.

 

 

 

8 tenths add 6 tenths makes 14 tenths, or 1 whole and 4   tenths. The 1 whole is 'carried' into the units column and the 4 tenths is   written in the tenths column.

 

 

Extend to numbers with any number of digits and decimals   with 1 and 2 decimal places.

 

 

 

Before   they use a written method to add decimal numbers, children should estimate the answer.

 

For   example, they calculate 13.86 + 9.481, and use rounding to check   that their answer is approximately 23, rounding to check that their answer is   approximately 23

 

(      Year 5/6)

Please remember that the ‘goal’ is   not to teach this method but to make sure the children have a very secure   grasp of the number system and mental strategies which will in turn allow   them to answer any addition problem without using this method. Once they have   that then they can choose which method to use.

 

 

 

 

 

 

 

 

 

 

 

 

 

Pencil   and paper procedures

 

      

 

 

 

 

 

 

 

 


    

 

 

    

 

 

Written methods for subtraction of whole numbers

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 supported by informal calculation strategies. These notes show the stages in building up to using an efficient method for subtraction of two-digit and three-digit whole numbers .

 

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.

Stage 1

 

In the early stages, children will be taught to ‘take away’ one or two objects and   find the new total.

 

 

For example: 5-3=2

                          

5 take 2 away is 3

              

 

Stage 2

 

The next stage is for children to be able to work out one less or several less on a number   line

For example: 6 – 1 = 5

  - 1

                                  

                      
      
      

 

      
      

                           

                      
      
      

0     1     2      3             4   

      
      

                     

                      
      
      

         5      6      7      8        

      
      

                     

                      
      
      

 

      
      

                   

 

 

Stage 3

 

At an early stage children are introduced to the concept   of difference and that subtraction can be worked out by counting on the   difference.

For   example: How much longer is this row of cubes than this one?

9 – 3   = 6

 

                                         
   

 

    		   

 

    		   

 

    		   

 

    		   

 

    		   

 

    		   

 

    		   

 

    		   

 

   

                                             +1   +1     +1

 

                             
   

 

    		   

 

    		   

 

    		   

 

    		   

 

    		   

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stage 4

 

Using the empty number   line

 

The empty number line   helps to record or explain the steps in mental subtraction. A calculation   like 74 – 27 can be recorded by counting back 27 from 74 to reach   47. The empty number line is also a useful way of modelling processes such as   bridging through a multiple of ten.

(The steps can also be recorded by counting up from the smaller to the larger number to find   the difference, for example by counting up from 27 to 74 in steps totalling   47.)

With practice, children   will need to record less information and decide whether to count back or   forward.

 

 

 

 

The counting up method

 

The mental method of counting up from the smaller to the   larger number can be recorded using either number lines or vertically in   columns.

The number of rows (or   steps) can be reduced by combining steps as they become more confident.

 

(      Year 2/3)

 

Steps   in subtraction can be recorded on a number line. The steps often bridge   through a multiple of 10.

15 – 7 = 8

        

74 – 27 = 47   worked by counting back:

        

The   steps may be recorded in a different order:

        

or   combined:

        

 

            

or:

              

With three-digit numbers   the number of steps can again be reduced, provided that children are able to   work out answers to calculations such as 178 + £ = 200 and 200 + £ = 326 mentally.

The most compact form of   recording remains reasonably efficient.

 

 

            

or:

            

  •   The method can be used with decimals where no more than three columns   are required. However, it becomes less efficient when more than three columns   are needed.
  •   This counting-up method can be a useful alternative for children whose   progress is slow, whose mental and written calculation skills are weak and   whose projected attainment at the end of Key Stage 2 is towards the lower end   of level 4.

 

 

 

 

 

 

 

 

 

            

or:

            

 

 

 

 

Stage 6

 

 

Decomposition should only   be taught when the children are a secure level 4 and not before year 5. They   can be taught this if they haven’t reached this level after SATs to support   their transfer to High School.

 

 

    


Written methods for multiplication of whole numbers

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 supported by informal calculation strategies. These notes show the stages in building up to using an efficient method for subtraction of two-digit and three-digit whole numbers .

 

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 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

 

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

 

.

Stage 1

First children are taught   to count in 2’s, 10’s and 5’s using practical objects.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For example:

 

3 x 2 = 6

 

     +        +          

 

3 x 5 = 15

 

      +      +      = 15

Stage 2

 

 

 Number Line Jumps

 

 

                                  

                      
      
      

 

      
      

                                                                                                                                           

                      
      
      

0     1     2      3      4          

      
      

                     

                      
      
      

         5      6      7      8

      
      

                     

                      
      
      

 

      
      

                          

 

 4 + 4 = 8

2 x 4 = 8

Or 4 x 2 = 2 +   2 + 2 + 2 + 2

Number line   jumps support mental methods

                                  

                      
      
      

 

      
      

                                                                                                                                           

                      
      
      

0             6            12       

      
      

                     

                      
      
      

           18          24            

      
      

                     

                      
      
      

 

      
      

                   

 

13 x 6 = 78

 

 

 

 

Stage 3:

 

Arrays

 

Arrays &   Repeated addition

 X                4                                                                   

l  l    l    l              4 x 2   = 2 + 2+ 2 + 2 = 8

l  l    l    l

                                              2 x 4 = 4 + 4 = 8

 

X                     5

          l    l    l    l    l      5 x 3  =15

   3     l    l    l    l    l      3 x 5 = 15

          l    l    l    l    l      15 ÷ 3 = 5

                                                15 ÷ 5 = 3 

 

Stage 4

 

 Partitioning units and   tens, hundreds  or thousands

 

 

Mental methods for   multiplying TU × U can be based on the distributive law of   multiplication over addition. This allows the tens and ones to be multiplied   separately to form partial products. These are then added to find the total   product. Either the tens or the ones can be multiplied first but it is more   common to start with the tens.

 

 

43   x 6   =             40 x 6 = 240

                             3 x 6 =    18   

                                          258

 

 

 

 

 

 

Stage 5

 

Arrays to grid

(      Year 2 - 3)

 

Arrays to grid 12 x 3

 

X                   10                                                       2

          l    l    l    l    l    l    l    l    l    l       l  l

   3       l    l    l    l    l    l    l    l    l    l       l  l   

          l    l    l    l    l    l    l    l    l    l       l  l

 

 


     x    10        2          

     3     30     6

     

    =    30 + 6 = 36

 

 

 

 

 

 

Stage 6

 

The grid method

 

As a staging post, an   expanded method which uses a grid can be used. This is based on the   distributive law and links directly to the mental method. It is an   alternative way of recording the same steps.

  •   It is better to place the number with the most digits in the left-hand   column of the grid so that it is easier to add the partial products.

 

(      Year 4 - 5)

(    (By the end of Year 4, children will be able to use an   efficient method for two digit by one digit multiplication)

38 × 7 = (30 × 7) + (8 × 7) = 210 + 56 = 266

                   

 

                                  
   

x

    		   

30

    		   

8

   
   

7

    		   

210

    		   

56

   

 

210   + 56 =200 + 60 + 6 = 266

 

Or                       210

                             56

                           200

                             60

                               6

                           266

 

Stage 7

Two-digit by two-digit products

 

Extend to TU × TU,   asking children to estimate first.

  •   Start with the grid method. The partial products in each row are added, and   then the two sums at the end of each row are added to find the total product.

(By   the end of Year 5, children will be able to use an efficient method for   two digit by two digit multiplication)

 

 

56 × 27 is approximately   60 × 30 = 1800. 

 

                                                    
   

x

    		   

  20

    		   

  7

    		   

 

   
   

50

    		   

              1000

    		   

350

    		   

 

   
   

 6

    		   

120

  &