This is a classic problem and is a lot of fun for this age group (and older children, who could move through it at a faster pace).
The basic problem is this: if you have a room full of people and everyone shakes hands, how many total handshakes are there? Some basic ground rules (that are good to have the children come up with): you don't shake hands with yourself; you only shake hands once with another student.
We did this with 1st, 2nd and 3rd graders. We demonstrated with 2 people, then 3 people, then 4 people and kept track of the number of handshakes. They split into groups and tried to figure out how many handshakes there would be if all 20 of them shook hands. Encourage them to write it out in a table, to try it themselves (lots of noise with that, but fun) and to look for patterns. The table should contain this information:
2 people, 1 handshake
3 people, 2 handshakes
4 people, 6 handshakes
5 people, 10 handshakes
6 people, 15 handshakes
Eventually they see that every time a new person joins the group, that person has to shake hands with everyone else in the group. So when the group goes from 6 people (with 15 handshakes) to 7 people, that 7th person has to shake hands with 6 people. The new total is 21.
The pattern they should see is that if there are n people in the group, then the number of handshakes is
1 + 2 + 3 + 4 + 5 +... + (n-1).
We went through it again to make sure we believed it for our table (write 15 = 1 + 2 + 3 + 4 + 5) and then they had a challenge: what if 100 people shook hands. How many handshakes would there be?
The students who see the pattern will know that it should be
1 + 2 + 3 + 4 + ... + 97 + 98 + 99
but they are not sure how to add all of those numbers. Again we go back to an easier example, say 10 people, and use grouping to do the addition. Ideally they should pair up the numbers so that they add to 10. So
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) +5 = 10 + 10+ 10+ 10 + 5 = 45.
Once they get this, they can work on adding the numbers 1 to 99. It takes awhile but they should see that it's 49(100) + 50=4950.
From here it can go several different ways. They could try to figure out how many handshakes with 1000 people using the same approach or they could try more examples. the idea is to get them to the pattern that makes it easy to add these numbers. That is, that
1 + 2 + 3 + 4 + 5 +... + (n-1) = n(n-1)/2.
This is a lot to do. It took us 3 weeks to go through all of this carefully (with a lot of reviewing each week) but they saw the connections and had a lot of fun with it.