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Why do this problem?

This problem offers students an opportunity to relate numerical ideas to real life contexts.
Thinking about different ways of counting the number of handshakes can lead to a better understanding of the general formula for triangle numbers.

Possible approach

This problem works very well in conjunction with Picturing Triangle Numbers and Mystic Rose. The whole class could work on all three problems together, or small groups could be allocated one of the three problems to work on, and then report back to the rest of the class.

Ask for seven volunteers to come and stand at the front of the class, and ask each volunteer to shake hands with everyone else, with the rest of the class counting how many handshakes take place. Was it easy to count? Would it be useful for the volunteers to shake hands in a more systematic manner? Repeat the process in the way suggested in the problem.

Allow some time for students to work out how many handshakes there would be with 8, 9 and 10 people, and discuss answers and methods.

Now present Sam's method and Helen's method, and ask the class to judge which method gives the correct answer.
"What is wrong with the other method?"
"How could it be modified?"
Students could check that their revised method gives the same solutions as those found above for 8, 9 and 10 people.

For a class that has been introduced to algebra, students could express "Sam's method" and "Helen's method" algebraically.

Finally, ask them to work out whether the following numbers could be the number of handshakes at a mathematical gathering, and how large those gatherings would be:
  • 4851
  • 6214
  • 3655
  • 7626
  • 8656

Key questions

What is special about the numbers of handshakes at different sized gatherings?
How does each method for working out the total number of handshakes relate to different ways of describing what happens when everyone in a room shakes hands?

Possible extension

Can you have a gathering with 9, 19, 29, 39, ... handshakes? Are these impossible? How do you know? What other impossible families of gatherings can you find?

Could there ever be a gathering with a multiple of 1000 handshakes? Give some examples.

Possible support

Students could represent people shaking hands by joining points around a circle. Circle templates with dots evenly spaced on the circumference can be found here.