In the last post we looked at adding single-digit numbers, such as 4+5 (which adds to 9, and is thus less than than 10), and 7+8 (which sums to 15, being larger than 10). I explained that these two types of single-digit addition are a core element of mental arithmetic.
Today I extend this to include two-digit numbers, such as 11+32 (which sum to less than 100), and 45+78 (which has a sum larger than 100).
I reiterate the message from the first post in this series – that the calculator MUST NOT be the first point of help 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 do the job better than your own brain.
In many cases, using the calculator will take you MORE TIME than using your brain to get the result. Developing your brain to do these calculations yourself is a vital part of learning mathematics.
Mental arithmetic is partly concerned with memorising 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 be able to do these mental operations quickly and accurately as part of more advanced mathematics.
In the last post I covered the addition of single-digit numbers, and I now extend this to two-digit numbers, which range from 10 to 99, and how the knowledge of single-digit addition will now help with 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 memorised these as facts which you can draw on as you need them. Without this skill to add single-digit numbers you will immediately struggle when using this for two-digit additions.
The general problem of addition is:
where and are both numbers from the range from 0 to 99.
Analysis of a Two-Digit Sum
Let us analyse two different sums of this form.
The first is
and the second is
The first difference between these two 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 we have memorised these already (you have, haven’t you!).
So in the second problem, the sum of the last two digits is 12, being 10 + 2.
Whereas this was previously referred to as “carrying” the tens digit, this term is not longer used since it is too focused on the method, rather than on the understanding of the place-value system which leads to these numbers. The focus in these articles is on the learning and memorising of these calculations mentally, so that that are imprinted into your memory and can be recalled at will. So whereas you may spend some time understanding these sums, it is essential that these are committed to memory as much as possible and so that you do not need to perform the sum every time you need these results.
The knowledge of adding numbers up to 100 is required to be learned by Grade 4, and will form a key element of knowledge from this point on. These should be both memorised for some cases, and for others they should be able to be worked out quickly and correctly.
I first want to 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 Mental Arithmetic Series document below, print it out, and keep this handy at all times until this is mastered.
This is only for the sums up to and for others you should work these out yourselves.
I now provide a number of worksheets, of increasing complexity, which include not only the number up to 20 + 20, but also beyond these.
Here are four worksheets, each with 10 questions. and these should be addressed in sequence.
For Grades 1-3 these are not expected to be done fast, but rather carefully, and memorised as much as possible.
From Grades 4 onwards these must be part of a memorised structure.
During the next Mental Arithmetic post I will be focusing on breaking these small numbers – up to 20 – down into their parts and exploring number bonds.
So….enjoy maths, and “MAKE YOURSELF COUNT”.
Email me if you have any suggestions and hints.