This is a small part of CFT by Dr Paul Gilbert that I think you could very usefully use with your children when they have had some kind of argument or fall out with someone. It works by acknowledging that when you are in different emotional states, your behaviour, thoughts and the memories you bring to mind are very much influenced by these emotional states. For the sake of this helping your children to consider a situation, the states to consider are: angry, anxious, sad and lastly, compassionate.
So say for example, your child has fallen out with someone because they had said, 'you are a really nasty person.' Ask you child to think of how angry they are and ask them what their angry self is thinking, how it is remembering the situation and how it wants to behave. So then ask the same questions of the anxious (worrying) self. Next ask what the sad part would be thinking, recalling and likely to behave. Finally ask the same questions of your child’s compassionate state. So as a simplified example, the angry part might hate the other child and possibly want to hurt her, the anxious part might be worried that others also think he is nasty and that others might get involved, the sad part might be wondering if what they said was true and the compassionate part would wonder if their friend was mean because they are struggling with something and want to help them.
Now clearly ‘compassion’ needs some explaining to most children. You can describe it as the part that is in everyone, that wishes to understand, help and stop the suffering of others. It involves putting the other person in a position of just as much (if not more) importance as ourselves and this can be hard for some children, especially if they tend to be most comfortable being angry. To access our compassionate self, Gilbert suggests we start simply by taking deeper and longer breaths to trigger our parasympathetic system. (e.g. breathe in for the count of six and then breathe out for the count of six.) In the past I have accessed this by meditating upon all the good things about the relationship which is suffering and imagining warmth between myself and the other person. When your child is in this state they will find they will be able to see the situation from different angles and have much more resource with which to tackle the issue affecting the relationship.