Mental strategies for addition and subraction

Most formal written algorithms for addition and subtraction work from right to left, starting with the units. In mental calculations it is much more common to work from left to right. This makes more sense, because you deal with the biggest and most significant bits of the numbers first.

One strategy is mentally to break the numbers up into hundreds, tens and ones, and then to combine them bit by bit, starting with the hundreds. So, for example, given 459 + 347, we would think of the 459 as (400 + 50 + 9) and the 347 as (300 + 40 + 7). This process is sometimes called partitioning into hundreds, tens and ones. We would then use the freedom granted to us by the associative and commutative laws to add these bits in any order we like. The ‘front-end approach’ would deal with the hundreds first (400 + 300 = 700), then the tens (50 + 40 = 90, making 790 so far), then the ones (for example, 790 + 9 = 799; 799 + 7 = 799 + 1 + 6 = 806). Notice that I have used 800 as a stepping stone for the last step here.
Writing this out in full, in a way which might explain my thinking to someone else:

459 + 347 = (400 + 50 + 9) + (300 + 40 + 7)

= (400 + 300) + (50 + 40) + (9 + 7)

= 700 + 90 + 9 + 7

= 799 + 7 = 799 + 1 + 6 = 800 + 6 = 806

We will quite often use the front-end approach to get us started in a subtraction done mentally. For example, for 645 – 239, we would immediately deal with the hundreds (600 – 200 = 400) leaving us simply to think about 45 – 39. This gives us 6 (using 40 as a stepping stone), so the answer is 400 + 6 = 406.