BOGTF - learning number bonds and other methods

I want to share some ideas about supporting Key Stage 2 learners with their mathematical development. The methods that I outline below are not necessarily recommended, and I accept that there will be mathematicians who will disagree with my methods. However, they have developed over the best part of a decade where I was the teacher who had the responsibility for ensuring that children who had 'failed' to make progress during Key Stage 2 caught up. Whilst this was a part of the "Standards game" my motivation was for the children, as not having a basic level of numeracy is as detrimental as in life as not being able to read. Shall I jump off my soap box?

Buy One Get Three Free
This method plays on the BOGOF offers that we are all so familiar with in the shops, except it goes one better - you get three free not two!

Purpose: To support the learning and recall of number bonds for pairs of numbers to 10, 100 and 1000.
Apparatus: It is useful for some learners to have some concrete apparatus for this activity.

• ten 1p coins or
• ten uni-link cubes or ten multilink cubes or
• ten coloured counters or
• ten sweets (Healthy Schools alert), etc. etc.

Method: For children that are really struggling to retain and recall the pairs of numbers that make ten it is advisable to play with the ten objects and investigate and explore different pairings. These could be recorded visually, but should be supported by writing as an equation.

Taking the example in the picture on the left, it can be seen that 4 + 6 = 10. Ask the pupils to reverse the coins, thus making 6 + 4 = 10. So here is the first free fact.

Knowing that there are ten coins, remove four of the coins. This is equivalent to 10 - 4 = 6. Replace the coins, making 10 once more. Now remove 6 coins, equivalent to 10 - 6 = 4.

Make sure that the pupil has understood what is happening through the use of the apparatus before writing the four equations:

4 + 6 = 10
6 + 4 = 10
10 - 4 = 6
10 - 6 = 4

From this point guide the pupils to find any common features of the four equations. Of course, you are looking for them to identify that each of the four equations has the same three numbers arranged differently. If the pupils don't see this easily, explore a different pair of numbers through the apparatus.

So, once the pupils have made the connection between the four equations you can move on.

Stage 2
Next, make a list of number pairs that make 10:
0+10, 1+9, 2+8, 3+7, 4+6, 5+5, 6+4, 7+3, 8+2, 9+1, 10+0.
If the pupils have understood the principles in the equation forming above they should notice that 5 out of 11 of the pairs above are reversals, and therefore to know all eleven pairs they only need to remember 6 of them (I think it is important not to forget the 0+10 pairing).

The benefit of the BOGTF approach comes out here. When working with pupils who have made very slow progress in Key Stage 2 I have found that one of the thing impeding progress is their self-esteem, and that this has a close link with a fear of having to remember so many things. By learning the five pairings of numbers that make ten, the pupils also have access to another 16 facts. (whoa - that should be 18 I hear you say! It is, but 5+5=10, 5+5=10 and 10-5=5, 10-5=5 reduces to two facts due to the repetition - linking with reducing to lowest common denominator much later in the progression!)

This realisation that if they learn 6 facts that they then have immediate access to 22 facts is an eye opener, and really empowers pupils. However, it is vital that they understand the earlier stage in this process. They have to be able to explain the equivalence (in their own way is fine) to be able to have an intellectual security in converting a known fact to one of its "free group".

Moving on to 100 and 1000
By applying knowledge of what happens when a number is multiplied by 10 the pupil can begin to understand that if they know 4 + 6 = 10 they can apply this to recall quickly that 40 + 60 = 100 etc. Again, where this is insecure, I would recommend using some visual stimulus or apparatus.