How many handshakes can people do?

2/5/2010 & 2/12/2010

The following problem gave kids introduction to arithmetic progression.

We had 11 kids in math class on that day.
I asked all kids to stand up and chose two kids to come forward. Then I asked everybody how many handshakes these two kids can do (condition, they should not do more than 1 handshake among each other). Kids said: 1. Two kids did 1 handshake. Correct!
Then I chose another child to join the first two and asked the same question. Kids calculated: 3.
3 kids did 3 handshakes. Correct!
I added fourth child asking the same question. Kids calculated: 6. 4 kids did their handshakes and confirmed that the answer is 6.
We did the same by adding more and more kids to the group.
At the point when there were 8 kids I showed them that by adding 1 more child the number of handshakes = Previous Number of Handshakes + The handshakes that the new kid has to do. So in case of 8 kids we have 21 + 7 = 28.
When we have 9 kids there will be 28 (from previous time) + 8 = 36. 10 kids: 36 + 9 = 45.

Then I showed that for
2 children -> 1 handshake
3 children -> 1 + 2 handshakes
4 children -> 1+2+3 handshakes
etc.

Next, I showed them arithmetic progression
1+2+3+4+5+6+7+8+9+10
I asked them a question whether they see a pattern if you add the first and last number, 1+10, second and second from last, 2+9,etc. All those sums are equal to 11.
Then I asked how many sums like this (or pairs) there are. They calculated: 5.
So we need to add 11 five times = >11 x 5 = 55
Then I asked kids to add numbers 1+2+3+...+10 and they got 55 as well.

I showed them that this rule works for any sequence (arithmetic progression). They tried to add 2+3+4+5, 1+2+3+4+5+6, etc. using the rule above (adding first and last number and multiplying it by number of pairs).