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This problem offers students an opportunity to relate numerical ideas to real life contexts.
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?"
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:
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.
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...
Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?