In the last post on Mental Arithmetic I explored the mental arithmetic for adding numbers up to 2-digit number. I now extend this to the topic of “number bonds” – which are various ways in which we can divide up a number into its parts and combine these parts back into the number.
This is a very important part of early mathematics development of learners, and it is essential that this part of mathematics is learned so that is becomes a habit which is imprinted into the mind of the learner, rather than requiring the learner to use a procedure for counting, since such procedures will slow down the learner.
For example, consider the sum . A learner may adopt an approach to first consider , which is easy. From this they may determine , followed by , and finally . This is only one of many possible approaches which a learner may use if they do not know the answer immediately. My concern is not whether this is the correct or best procedure to be using, but rather I am concerned with the usage of any procedure at all, rather than having this basic knowledge memorised and imprinted mentally so that it can be recalled immediately without any need for calculation.
Within the Foundation Phase, Grades R-3, there is much emphasis on building up number bonds, which are ways in which numbers combine to make up a given number. For example, divide the number 10 into two smaller numbers, and you can get the following:
In addition, there are also two others which need to be mentioned:
These are presented as a sum with an answer, and another approach is to ask learners to take a number and to splitit up into parts in different ways. For example, “split up 12 into two other numbers”, for which the learners should respond with or . Thus mathematics is not only combining numbers, but it also concerned with splitting up numbers in different ways.
Throughout Grades R-3 these number bonds are worked out with many examples and using objects to help to identify these sums, such as the use of different shapes and physical items, such as matches or sticks. All of this is for the purpose of ensuring that this knowledge becomes imprinted into the learner’s mind – so that the learner can recall these immediately without having to add up the sums every time that this is needed. In my 40+ years of experience as a mathematics tutor I continually 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 we will address later in this series, there are a range of simple arithmetic calculations which should be memorised, so that these can be used as part of larger and more complex calculations which are covered throughout all of further mathematics. At this point in these posts on Mental Arithmetic I am building the basis for efficient calculation, and part of this is working our what should be recalled immediately, what should done mentally or on paper and pencil, and finally when it is better 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 memorised 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, the calculations should be second nature and the learners should be able to recall immediately all of these types of sums and without having to waste time in performing the calculations mentally – or using paper and pencil or a calculator.
Number bonds for 10
The list above already shows the number bonds up to 10, so these are not repeated here.
Now you should try out some to see if you can work these out yourself, and one of the best ways is to “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 give you the answer and you come up with the question. 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 , or .
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 number bonds for this number. For example, let us take the number 60, which is part of the knowledge needed for the understanding of minutes and seconds when measuring time.
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