### Number bonds

I have searched the internet, read numerous research articles and found plenty of tasks designed to help kids gain a deeper understanding of number bonds. This post contains a brief explanation of number bonds and why it is so important that kids really understand them. I have also included a list (with descriptions) of some of my favourite tasks, along with a video example of this learning in action.

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What are number bonds?

Number bonds, also know as addition facts, are pairs of numbers that make up a given number. Young primary school children are expected to learn number bonds to ten and then twenty. As they progress through school, they will also learn number bonds to one hundred and one thousand.

Children are not only expected to be able to understand number bonds but should be able to recall them almost instantly. They should also be able to confidently use the corresponding subtraction facts. For example, for number bonds to ten:

0 + 10 = 10           10 - 10 = 0
1 + 9 = 10           10 - 9 = 1
2 + 8 = 10           10 - 8 = 2
3 + 7 = 10           10 - 7 = 3
4 + 6 = 10           10 - 6 = 4
5 + 5 = 10           10 - 5 = 5
6 + 4 = 10           10 - 4 = 6
7 + 3 = 10           10 - 3 = 7
8 + 2 = 10           10 - 2 = 8
9 + 1 = 10           10 - 1 = 9
10 + 0 = 10           10 - 0 = 10

Kids should also be able to understand that the order in which you add the numbers does not matter (e.g. 2 + 8 is the same as 8 + 2).

Why is learning number bonds important?

The term 'number bonds' might seem like another unnecessary, abstract mathematical phrase. However, although the phrase has only been in use in UK schools since the 1990's, the concept is nothing new - it simply involves addition and subtraction and is vitally important.

The idea of learning number bonds is to build up a mental picture of the two quantities that together make the required number. This enables children to be able to do mental arithmetic. Particularly important for mental maths are number bonds to 10, as this links in to place value and our base 10 number system.

Knowing number bonds with confidence allows children to develop strategies for solving more complicated mental problems. As a simple example, to carry out the calculation 7 + 5 mentally, a child who can instantly recall their number bonds to 10 will know that they need to add 3 on to 7 to get up to 10. They then have 2 left over from the 5 and can now add that on to the 10 to make 12. Essentially, they are breaking the addition up into 7 + 3 + 2 = 10 + 2 = 12.

Tasks to help kids learn number bonds

The Kiducation UK video below demonstrates some of the tasks and ideas that follow. It shows how the child's understanding develops throughout the task and provides ideas for the type of questioning that can be used to enhance the child's learning.

When introducing children to number bonds, it is important not to worry about the formalities. It doesn’t matter that they are called 'number bonds' and children don’t immediately need to be able to repeat all of the required number bonds or recall them in the correct order. First, simply introduce them to the idea of splitting one quantity into two groups and let them discover for themselves how it can be done. They will inevitably start to notice the patterns for themselves after a while and then you can begin to formalise it for them. By letting them play and discover, they will really understand the meaning of a number bond and what it is they are doing. It will then be easier for them to extend this on to larger numbers in the future. It will also make memorising them easier and will allow for easier use of number bonds when attempting to solve mental arithmetic.

Investigate how many ways you can split 5 objects into 2 groups
Start by giving the children a small group of objects (Lego bricks, counters, sweets, toys…). I would suggest five, as this is the number that many children can recognise without having to continually recount them. Then simply ask them how many ways they can split the objects into two groups (you need to emphasise “two” here, as some children get very inventive and try to split them into three or more groups).

Develop strategies to find all the different possibilities and make a visual record
Ask them to discuss it and compare with their friends – if they don’t start to do it for themselves, you can then begin leading them towards strategies for formalising it. For example, start with groups of five and zero, and then move one object across at a time. Getting the kids to record their answers (visually rather than with numbers) then creates a full list of all the number bonds to five. This gives them a visual image of the number bonds and really emphasises that all the pairs of groups add up to five (as they do not change the total number of objects that they have). A kinaesthetic activity like this can help the students gain mental images of number bonds, helping them remember and use them in future.

Investigate number bonds to 10
Once the children understand the concept of number bonds (as splitting one number into two groups in lots of different ways) you can move them on to larger numbers. Again, use images and objects to start with. Although ten objects are harder to recognise without counting, the kids should now be able to understand that the total isn’t changing and should have developed strategies to find all the number bonds. To help kids that don't understand that the total remains as ten, use something concrete that they know is ten and they know doesn't change (their own fingers, or the ten beads on an abacus row). Use the task to create a visual resource that you can constantly refer back to when practising in the future.

Practice, practice, practice
Now the kids have discovered number bonds for themselves, really understand what they are all about and have developed mental images, it is time to practice, practice, practice. The only way for kids to be able to instantly recall number bonds is to come across them constantly, in multiple different situations and in various settings. This reduces the monotony and enables them to apply number bonds to different situations in the future. Consistently using the same example may lead students to think that number bonds only apply to that one example and are not useful in other situations.

There are many different examples that can be used for practising, and I have come across numerous fun, inventive and different ideas across the internet. Some of my favorites are:
• Play number bonds snap - snap, but the numbers must add up to 10 rather than being the same.
• Make a 10 sided dice with a twist - It is easy to find a net for a 10 sided dice on the internet (as in the picture). Get the kids to label the top five sides from 1-5. Then explain that opposite sides should add up to 10. The kids will get to practise number bonds whilst making the dice and then can use it to play games and practice some more (e.g. guess what number is on the bottom, play a board game using the number on the bottom rather than the top, ...)
• 10 make a treat - this is great for at home. Give the child a counter whenever they are good and a special treat when they have earnt ten. You can keep asking them how many more they need. Not only does this get them practising their number bonds, it also encourages good behaviour!
• Play pairs and other games - I found a lovely set of cards on TES resources that have the numbers represented visually and with numbers. The visual cards are perfect to help kids who are struggling to remember the number bonds and to reinforce what number bonds are really all about. The number cards are great for getting children that are more confident with their number bonds to keep improving.
• Match up activities - I found a lovely ice cream match up task that is great to help more confident children practise their number bonds using numbers rather than images.
• How many did I take away? - start with a group of ten objects and take some away without the children seeing. Ask the children to look at what is left and guess how many have been taken away. This is a great activity for helping children practise and understand the corresponding subtraction facts.
• Bowling - this is similar to the above task. After having a throw, ask the children "how many did you knock down?", "how many are left standing?".

Number bonds to 100
When extending children on to number bonds to one hundred, it is helpful to start by making the link to number bonds to ten and to place value, e.g. 4 + 6 = 10, 4(tens) + 6(tens) = 10(tens).

To move on to more difficult number bonds to one hundred, such as 72 + 28, a Slavonic abacus is a great resource. There is a fantastic article on TES that explains it in detail. A Slavonic abacus breaks up each row into fives by using two colours, and this helps kids to really visualise the number pairs and "see" the number bonds. Learning number bonds to 100 really helps mental arithmetic - I have had so many students make mistakes such as 62 + 48 = 100 (as they are recalling that 6 + 4 = 10).

So much more than just number bonds

Teaching/learning number bonds might seem repetitive and tedious, but there are so many opportunities to use the investigation of number bonds to learn about and discover many other mathematical concepts. For example:
• The tasks using objects help children to understand numbers as quantities, as a collection of 'ones' rather than a sequence of words.
• It helps children grasp the concept of adding as combining two groups (or quantities) and subtracting as taking one away from another.
• It helps children to make the link between adding and subtracting and understand the relationship. All of the suggested activities can be used to talk about subtraction, and constant reinforcement will help kids to visualise and understand that subtraction "undoes" addition.
• It helps children to see that addition is commutative (i.e. the order in which you add the numbers doesn't matter). By recognising that, for example, 7 + 3 and 3 + 7 both produce 10, kids will begin to understand that they can add numbers together in any order. This will help them in the future with mental maths (it is easier to add a smaller number to a larger) as well as more difficult concepts such as rearranging equations and simplifying algebra.
• The tasks can help develop children's investigative and problem solving skills. By asking them to find a way of finding all of the number bonds and recording the results systematically, children are developing strategies that they can use to help solve similar problems in the future.

I strongly believe that kids gaining an understanding of number bonds through discovery and investigative tasks can aid the learning of other mathematical topics in future. It is important to make sure that students really understand number bonds, their use and their relationship to numbers and other mathematical concepts. Learning numbers bonds simply as "number facts" that they must acquire through repetition will have very little benefit to the students and misses an opportunity to deepen their understanding of maths.