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.
Steps in addition, can be recorded on a number line. The steps often bridge through a multiple of 10.
The empty number line helps to record the steps on the way to calculating totals.
Stage 2- Partitioning
The next stage is to record the 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 are placed under tens. This also links to mental methods.
Stage 1- 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 totaling 47.
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 effiecient for the calculations such.
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 combing steps. With two-digit numbers, this requires children to be able to work out the answer to calculations such as 30 + ⮽ =74 mentally.
Stage 1- Mental multiplication using partitioning
Mental methods for multiplying TU x 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.
Children start by learning about halving and even numbers. They will understand equal groups and share actual items out in play and problem solving. Children will develop their understanding of division and use jotting to support calculations.
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.
Mental methods for dividing TU ÷ U can be based on partitioning and on the distributive law of division over addition. This allows a multiple of the divisor and the remaining number to be divided separately. The results are then added to find the total quotient.
Children should also be able to find a remainder mentally, for example the remainder when 15 is divided by 2.