In the last post I outlined the importance of having a well-developed proficiency in mental arithmetic to help you through school and in real-life work.
Today I examine the first step of mental arithmetic knowledge, which is simply to know the sum of two single-digit numbers, being the numbers are 0-9. When two are added they create a number in the range 0-18.
I reiterate the message from the first post in this series – that the calculator should NOT be the first point of help when you are confronted with a problem, and should only be used when it is necessary. In many cases, the calculator will take you LONGER to use than using your head to solve a problem, and there are host of problems for which the calculator is not useful, since it fails to identify the steps which you may need to show in your answer to a problem.
Whereas this problem of adding two single-digit numbers may seem trivial, it has been my experience with learners of all grades that even this primitive knowledge is not fully memorised, and thus I start my venture into mental arithmetic here, and I apologise in advance for those who are experienced and for whom this is a waste of time.
Whenever are adding numbers of any size, we break down the larger problem of adding larger numbers down to the small problems of adding individual digits, and thus this important and primitive knowledge to help us to address larger problems. To become proficient in more complex problems, we need to add single-digit numbers quickly and efficiently.
The general problem is
where and are digits from the range 0-9, being 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Let us divide these sums into three groups:
- firstly, those which add to a number which is less than 10 – such as
- secondly, those which add up exactly to 10 – such as
- thirdly, those which are larger than 10 – such as
The second of these three groups is important in another context, finding which numbers can add up to a specific number, which in this case is 10, but we would equally choose any other number, such as 7 or 12. This is called “number bonds” and will be addressed in a later post.
The knowledge of adding numbers up to 10 is required to be learned by Grade 1, and the knowledge of adding up to 20 by Grade 2, and these must be memorised. It is not acceptable for senior learners not to have learned these simple calculations, since then they cannot recall these at will and will waste a lot of time in performing calculations which they should have memorised. These simple additions are fundamental to all of arithmetic, and occur in all calculations involving numbers.
First – download the Mental Arithmetic Series document below, print it out, and keep this handy at all times until this is mastered.
Second – download the following exercises, and work through each of these until these are embedded into the knowledge. Always correct errors immediately, and then redo the correct answer a number of times to ensure that this is remembered properly, since it is quite easy to remember the incorrect answer!
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. From Grades 4 onwards these must be part of a memorised structure, and from Grades 7 onward these must be solved in 15 seconds or less – for all 10! As fast as you can write you should be putting down the correct answer.
If you do want more than write to me and I will gladly create and upload many more of these for you.
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.