In the last article in this series on Mental Arithmetic we considered the problem of adding double-digit numbers.
The scope of mental arithmetic is now extended this to the topic of “number bonds” – which are the various ways in which a number can be divided up into two parts which then can be added together to get back to the number.
So number bonds are really the opposite of addition, since in addition we add two numbers to get a single number, and with number bonds we take a single number and then divide it into two smaller numbers.
This is an important part of early mathematics development and it is essential that this is imprinted into your brain as facts, so that you do not need to use a procedure for counting, since procedures take more brain work and slow you down.
For example, consider the sum . You may use an approach by first looking at the smaller problem , which is trivial. From this results you may look at , followed by , and finally . This is one of many possible approaches which you may use to find the sum if you have not memorized this. My concern is not whether you are using the correct or best procedure, but rather why you are using a procedure at all rather than the memorization of this so that it can be recalled immediately without any need for calculation procedures.
Within the Foundation Phase (Grades R-3) emphasis is placed on building up number bonds, which are ways in which smaller numbers combine to make up a larger number. For example, consider how you can divide the number 10 into two smaller numbers, and you have the following ways you can do this:
In addition to these examples, there are two others which I need to mention:
These are presented here as a sum with an answer, and another approach is to ask learners to split a number up into two parts in different ways. For example, consider the question “split up 12 into two smaller numbers“, for which you may respond with or , both of which are correct. Mathematics is used not only to combine numbers by adding them together, but also concerned with splitting up numbers in different ways and these number bonds will be used for us to explore subtraction in the next article.
Within Grades R-3 these number bonds are worked out with many examples and by using physical objects, such as matches or sticks, to identify these bonds. This is for the purpose of ensuring that these facts about the number bonds are memorized and imprinted into the learners’ minds – so that the learner can recall these immediately without having to add up the sums every time that this is needed. In my 50+ years of experience as a mathematics tutor I often encounter senior students who are unable to immediately recall the correct answer to simple sums, and who use their fingers to count, or take some time as they mentally compute the numbers. Many also reach out for their calculator for all problems, no matter how simple.
Whereas there is much emphasis on the “times table” for multiplication, which is coming later in this series, there are many other types of arithmetic calculations which should be memorized, so that these can be used as part of larger and more complex calculations which are covered throughout all of further mathematics. In these initial articles on Mental Arithmetic I am building the basis for efficient calculation, and part of this is working out what should be recalled immediately from memory; what should done mentally or on paper and pencil; and when it is necessary to use a calculator.
In Grade 1 the learners are expected to work with number bonds up to 10; for Grade 2 up to 20, and for Grade 3 up to 30. During this process the learner will become more familiar with these number relationships and this will result in this knowledge becoming more imprinted. However, it is also important that the RIGHT knowledge is imprinted, since when WRONG answers are memorized these are quite difficult to unlearn and to forget.
In Grades 4-6 this knowledge is repeated, and is extended up to multiples of 10, 100, 1000. So by the time the learner enters Grade 7, these calculations should be second nature and the learners should be able to recall immediately all types of sums as facts without having to waste time in performing the calculations mentally, in using paper and pencil, or by using a calculator.
Number bonds for 10
Above I showed the case of number bonds up to 10, and you should refer to these and memorize these.
Now you should “fill in the blank” in some simple exercises, such as the following:
These are all at the Grade 1 level, and should be easy for those in Grades 2 onwards. If you are struggling with these then you MUST practice these to ensure that you know all of these and do not need to count on your fingers to get the answer.
The use of the symbol is important, since this represents the missing value, or the unknown quantity, and when algebra is introduced in further grades, this symbol is replaced by a single letter such as , so that will be written as .
Number bonds to 20
This is similar to finding the number bonds for 10, and this is something which you can try yourself.
Given the number 20 – find two numbers which add up to 20.
This is a “reverse” question, in which I have given you the answer and you must come up with the question in the form of “Add 16 + 4”. Whereas this may seem like a nonsense question, it provides you with the opportunity to be creative – since there is no single correct response and any response which is correct will satisfy the answer. For example and are both correct answers.
The same approach as used in the number bonds for 10 can also be used for 20. Try the following
Number bonds to 30
These are included into the curriculum, but are less useful in practice, and these rather represent a useful exercise for different number systems and to develop a sense of the number bonds which will apply to any number as the sum.
Number bonds to 100
This is a very common requirement, and should also be fully known by the time that the learner has moved through Grades 4-6. This situation occurs extensively in money management, and especially in terms of working out how much change. For example
I purchase items which cost R 4.25 and I present a R 5 coin. How much change should I receive?
However, in South Africa at present, there has been a removal of the 1c, 2c, and 5c coins, so that 10c is now the smallest denomination, so these examples are less important than they were in previous times when 1c, and 2c were part of the currency structure. However, for bank calculations, such as interest on borrowings and savings, these remain significant.
Other number bonds
To take this knowledge further, select any number, and create the number bonds for this number. For example, take the number 60, which is part of the knowledge needed for the understanding of minutes and seconds when measuring time and use this to build number bonds which add to 60.
Let us take a simple question which we need to solve:
How many minutes are left in one hour if 23 minutes have already passed?
This should be imprinted as an important part of mental arithmetic, since you will be doing this a lot in both mathematics and in your every day life with working with time (which is also mathematical).
Try the following:
Try it yourself
It is important to become familiar with these number bonds on the more common numbers which occur in both mathematics and in real life.
These include the base numbers 10, 12 (for dozen), 20, 60, and 100 as a minimum.
For each of these base numbers you should be able to to the following:
- given a base number, fine two numbers which add to this base number
- given one number, find a second number which adds to a base numbers
- given two numbers, determine whether they add to the base number or not