In the last article you added single-digit numbers, such as

- 4+5 (which adds to 9, a single-digit answer), and
- 7+8 (which sums to 15, being larger than 10, a two-digit answer).

I mention again that these two types of single-digit addition are at the core of mental mathematics.

In this article, I extend this to the addition to double-digit numbers, such as 11+32 (which sums to less than 100), and 45+78 (which has a value larger than 100).

I repeat the same message from the first article in this series – that y*ou must not reach for the calculator when you have a new calculation to perform*. Rather, you must only use the calculator when it is the best tool for the job, and can do the job better than your own brain and its stored facts.

Developing your brain to do these calculations yourself is a vital part of learning mathematics since using the calculator will take longer than using your stored facts in many cases.

Mental arithmetic is partly concerned with memorizing basic arithmetic facts and routines, and is also partly the basis on which many types of simple problems can be solved quickly. This becomes important as you move up the grades in school, since you are expected to do these mental operations quickly and accurately as part of more advanced mathematics. These will be used throughout all mathematics which you do in the future.

In the last article you worked on the addition of single-digit numbers, and I now extend this to double-digit numbers, which range from 10 to 99, and how the knowledge of single-digit addition will now help you with these more complex problems.

However, before you continue with this, ensure that you have mastered all of the single-digit addition problems, and that you have memorized these as facts which you can draw on as you need them. Without knowing the facts of single-digit addition you will struggle when using these facts to help you with two-digit addition problems.

The general problem of addition is:

where and are both numbers in the range 0 to 99.

## Analysis of a Double-Digit Sum

Let us analyse two different categories of double-digit sums.

The first sum is

and the second sum is

The first difference between these two sums is in the last digit of each of the two-digit numbers, 23 and 45, and I want you to focus on this last digit – the “*units*” digit.

For the first sum these units digits are 3 and 5, for which the sum is 8, and for the second these digits are 3 and 9, for which the sum is 12.

So we see that looking only at the last digit in a large sum, we have the same problem which we have dealt within the previous article in this series, being the sum of two single-digit numbers.

Before considering anything more about these numbers, we can already determine the sum of the last two digits, since you have memorized as facts and which you can now call on from your memory.

In the second problem, the sum of the last two digits is 12, being 10 + 2.

This is sometimes referred to as “*carrying*” the tens digit, although this term has gone out of fashion more recently. The focus in these articles is on the learning and memorizing of these calculations, so that that are imprinted into your memory and can be recalled quickly. So whereas you may spend time understanding these sums, it is essential that these are committed to memory as facts so that you do not perform the sum every time you need the value.

You should be able to add numbers up to 100 is required by Grade 4, and this will form a key element of your factual knowledge from then on. These should be both memorized for more simple cases, and for others they should be able to be worked out quickly and correctly as a mental operation.

We will first focus on the sums which involves numbers up to 20, so for two of these number there will be a sum which is a maximum of 40.

To get started download the document below, print it out, and keep this handy at all times until this is mastered.

Mental Arithmetic – Adding Double-Digit Numbers

This is only for the sums up to and for others you should work these out yourselves.

## Exercises

I now provide a number of worksheets, of increasing complexity, which include firstly sums up to 20 + 20, and then more challenging sums.

There are four worksheets, each with 10 questions, and these should be attempted in sequence.

For Grades 1-3 these are not expected to be done fast, but rather carefully, and memorized as much as possible.

From Grades 4 on these must be part of a memorized structure.

Practical – Double-Digit Addition #1

Practical – Double-Digit Addition #2

Practical – Double-Digit Addition #3

Practical – Double-Digit Addition #4

Email me if you have any suggestions and hints.