Why do this problem?
gives learners a great opportunity to apply number facts and skills to a seemingly spatial problem. It is also a good context for them to give reasons for the patterns which emerge.
Many could benefit from having sticks or straws with which to investigate this problem, although it can be tackled purely by sketching sticks.
What is the least number you may have in a "set of sticks"?
If you start with a set of $2$, how many sticks would be in the set placed across these $2$?
How many crossings would that make?
Can you think of a good way of recording what you have done?
Do you notice anything about the number of sticks in each set and the number of crossings?
How do these numbers relate to the total number of sticks?
If you have $10$ sticks how many could you put down to start with?
How many would be left to cross them?
How many crossings would this make?
Can you arrange them to make more/less crossings?
Can you generalise your ideas for any number?
Suggest using sticks or straws to model the questions asked and then sketch what has been done.