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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.

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

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?

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?

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.

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.