This idea works well as a first lesson with a new class. I have used it many times with primary pupils who are on a secondary school taster day, but you could adapt it for any age.
When pupils arrive, shake hands with some of them - they may find this funny. Then ask them: "If everyone in the room shakes hands with everyone else in the room, how many handshakes will there be? How can you convince us that your answer is right?"
There are several ways in which pupils may come up with the right answer. Then it gets more complicated. What if every possible group of three shakes hands together? How many ways are there of making groups of three out of all the people in the room? What about groups of four or any other number?
Colin Foster teaches maths at King Henry VIII School in Coventry.