students can introduce themselves to each other by shaking hands (or a high five or a 'namaste')
maybe start with 5 or 6 students at the front of the room and get them to all shake hands with each other
then think about how this can be done (maybe more) systematically - to make sure everyone has shaken hands with everyone else
"and how many handshakes was that altogether?"
think of a way or ways to record or represent what they did (seeking some variety)
various methods for recording 'handshakes' can be compared
if 5 people shake hands with the 4 other people this suggests there would be 20 handshakes
why do you half this?
the 'handshake' numbers are triangular numbers
but whereas the rule for the number of handshakes for 'n' people is n(n - 1)/2
the rule for the nth triangular number is n(n + 1)/2
the nrich site has a demonstration of how two triangular numbers fit together
this work will hopefully lead to an appreciation of a general rule which can be developed to consider how many handshakes there would be for e.g. the class, year, whole school and the locality
what digits do triangular numbers not end with?
why is that?
which two sum to 2500?
which sum to 10,000?
which triangular numbers are multiples of 11?
why is 1 + 2 + 3 + 4 a triangle?
how can odd 'n' triangular numbers be reformed into a rectangle?
the even 'n' triangular numbers?
the 'nrich' site has a very useful programme for drawing 'mystic' roses
as well as counting cans in growing stacks you can analyse how many cans touch 1 other, 2 others, 3 others etc.
you could explore stacks of 'cans' in 3 dimensions:
how many sweets?
- assuming that there are layers of sweets (which I don't think is the case)
triangular numbers occur when you select 2 things from a number, 'n'
you can also extend the considerations by looking at different ways to choose e.g. 3 from 5 (working towards 3 from 'n')