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 3- Expanded method in columns
Move onto a layout showing the addition of the ones to ones and the tens to tens separately. The total of the partial sums can be found by adding them - ones digit first.
The expanded method leads children to the more compact method so they understand its structure and efficiency.
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 number with different numbers of digits
Stage 2 - Partition
Subtraction can be recorded using partitioning to write equivalent calculations that can be carried mentally. This requires children to subtract a single-digit number or a multiple of 10 from a two-digit number mentally. The method of recording links to counting back on the number line.
Stage 3- Expanded layout, leading to the column method.
Partitioning the numbers into tens and ones and writing one und the other mirrors the column method, where ones are placed under ones and tens under tens.
This does not link directly to mental method of counting back or up but parallels the partitioning method for addition. It also relies on secure mental skills.
Stage 2- The grid method
As a staging post, an expanded method that uses a grid directly 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 step.
It is better to place the number with the most digit in the left-hand column of the grid so that it is easier to add the partial products.
Stage 3-Expanded short multiplication
The next step is to represent the method of recording in a column format, but showing the working. Draw attention to the links with the grid method above. Children should describe what they do by referring to the actual values of the digits in the column.
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.
Stage 2-Short division of TU ÷ U
'Short' division of TU ÷ U can be introduced as a more compact recording of the mental method of partitioning.
Short division of a two-digit number can be introduced to children who are confident with multiplication and division facts and with subtracting multiples of 10 mentally, and whose understanding of partitioning and place value is sound.