SandonBEC Numeracy Help

There are numerous ways to arrive at the correct answer for a calculation. Adults use different methods and therefore we must assume that not all children will use the same method. Any method may be used, even though it might be a non-standard method, providing the correct answer is achieved. In this section there are examples of methods that are often used in schools

Before starting a calculation pupils should be encouraged to work out an estimate of the answer by rounding the numbers involved to

Addition of single digits and two two digit numbers should be attempted mentally as often pencil and paper methods confuse rather than help. However, when using pencil and paper methods square paper often helps to maintain columns and ensure that there is only one digit in each column. The commonest mistake made by pupils is not being aware of where to place carried numbers.

In this example the tens digit from 7 + 4 has not been correctly carried correctly into the tens column

This calculation is correct as the tens digit has been added to 60 + 20

Mentally children will often add the largest numbers first and then go onto the units.

Some pupils may ‘bridge’ across the 10 by adding on 3 and then one in order to add on four. Also in this method the sum has been seen as 60 + 20 + 7 + 4, i.e. the tens and the units have been seen as separate entities or what is called ‘partioned’. Most importantly, place value is preserved, so that the 6 digit in 64, and the 2 digit in 27 are seen as 6 tens and 2 tens. It should be noted that mental methods are varied and will depend on the individual.

By rearranging the order it can be seen that 8 + 2 = 10 and so does 7 + 3. Therefore the answer is 20. This process illustrates the commutative law of addition (order does not matter).

Doubles or nearly doubles

Rounding numbers to the nearest 10

At primary schools children may not have encountered formal written methods until year 4. It is much more likely that they will have used number lines to help with computation.

In this case a simple straight line has been used to help as informal jottings to aid the thinking process. At school some children may need number lines with marks or numbers on. These are available from the Maths department. The way that people think about the addition process will differ from one person to the next, the above being just one example of many.

(Bridging means splitting the number up to go to the nearest 10 and then adding on the remainder. In this case the 4 has been split into 3, to reach 90 and then remaining one has been added).

In this case 27 has been rounded to 30, added on and then the ‘extra’ three has been subtracted.

In all cases the number line is used as a prop to aid thinking. The way that people think about the problem is individual to them.

Again mental subraction is often the easiest method when dealing with two digit and one digit numbers. There are some problems which positively benefit from mental calculation rather than pencil and paper methods.

e.g. The change from £10 when paying for an item worth £4.89. Using pencil and paper methods this would involve exchang ing. Instead it is easier to think of how many pennies to make up to £5 (11p) and then how many whole pounds are needed to make up to the £10 ( which would be £5) making the answer £5.11. It should be emphasised to pupils that a £ sign OR pence sign is used but never both together.

Problems such as 10 000 take away 1 become very difficult using pencil and paper methods. Subtraction can often be dealt with as ‘counting on ` or ‘counting back’.

There are various methods but the ‘ exchang ng method’ seems to be the most widely in use.

It can be seen that 7 can not be taken away from five and therefore ten must be exchang ed from the next column. Unfortunately some people would think that the answer to the above problem was 212, because they have failed to realise that 5 take away seven is not the same as seven take away five. This problem could have been solved mentally by doing 675 - 470 (467 to the nearest 10) and then adding three to adjust the answer.

Again, some people will benefit by using a number line to carry out subtractions. As with addition there are a number of different ways that a subtraction can be carried out.

In this case the person has ‘counted back’ from 67. Some pupils will favour the ‘counting on’ method, both mentally and with the aid of a number line.

These are only two examples of the ways in which people may use number lines to help with their thinking process when tackling subtraction problems, but there are many more.

Despite attempts by numerous teachers, both at primary and secondary school, there are still some people who do not know their multiplication tables. As the emphasis of the National Numeracy strategy is mental arithmetic there should be fewer people who go through life without these basic facts known.

e.g. 5 items at £5.99 each is better attempted as 5 x £6 subtract the extra 5p = £29.95.

If the calculation is set out in columns it becomes a calculation involving carrying.

For some people informal jottings will help pupils to work out multiplication calculations. Again number lines can help.

At Key Stage 2 (school years 3, 4, 5 and 6 pupils will practice doubling. Therefore to multiply by 2 they will double, to multiply by 4 they will double again and to multiply by 8 they will double again.

Multiplication of a two digit number by a one digit number can be done mentally.

People may have their own strategy for mental calculations. However, there are a number other ways in which multiplication can be carried out.

In this example multiplication is carried out as repeated addition, in the way that all multiplication can be done. Some pupils may add four lots of 35, again the number line is to aid thinking. This method is particularly useful for pupils who are able to carry out addition calculations but are weak on multiplication tables.

In this method the tens and units are ‘partioned’, calculated separately and added together at the end.

This would be a poor commentary on the calculation, as the digit 3 is not seen as 30. It is important to emphasise place value.

However, it is common for pupils to make errors using this method as they forget about writing in a zero when multiplying by a multiple of ten or a hundred.

The following method should be accessible to all children . The product (multiplicaton) of the number above and to the side is shown in the box.

You can then mentally add up the totals across and then add the two totals together, ie

Alternatively you could add each part separately. However in each case it is important that place value is preserved. Square paper often helps ensure that digits remain in the correct columns.

Everyone seems to find division difficult. Th e method which most adults will be familiar with is as follows:

However, at primary school pupils will have been taught about ‘grouping’ and ’chunking’

The answer is 4 remainder 1 as can be seen, The pupil has made 4 ‘groups’ of 3 and has 1 left over.

As informal jottings pupils will sometimes use lines to carry out the same calculation.

I I I

Four groups of 3 with a remainder of 1.

Alternatively, it could be thought of as ‘sharing’ 13 sweets between three pupils.

Although the result is the same, 4 remainder 1, it can be seen that grouping and sharing are different ways of looking at the same problem. In grouping there are 4 groups of 3 with a remainder of 1. In sharing you effectively end up with 3 groups of 4 with one left over.

In this method ‘counting on’ has been used. The number line can also be used to count back.

In this method division is being carried out as repeated subtraction. This method could be quite time consuming for large numbers. A more appropriate method for most calculations would be the refined stage of repeated subtraction, known as ‘chunking’.

Chunking is so called because instead of taking the devisor away as separate amounts, as in the above example, chunks of the devisor are taken away. The size of the chunks will depend on how competent the child is at multiplication.

The answer to the calculation is found by adding the left hand column of numbers

Long division seem to frighten children (and adults) because it looks so complicated and involves them having to do rough calculations. Many adults will be familiar with the ‘bus shelter’ method.

This method is poor because it is a procedure without understanding the true place value of the numbers. As with short division, children are more likely to be familiar with the ‘chunking, method.

This can be checked by doing 17 x 52. In all types of calculation you should make a rough estimate of the answer and check answers carefully.