median
don steward
mathematics teaching 10 ~ 16

## Sunday, 1 September 2013

### handshakes

a good task for the start of a new year
students can introduce themselves to each other by shaking hands (or a high five or a 'namaste')

maybe start with 5 or 6 students at the front of the room and get them to all shake hands with each other
then think about how this can be done (maybe more) systematically - to make sure everyone has shaken hands with everyone else
"and how many handshakes was that altogether?"
think of a way or ways to record or represent what they did (seeking some variety)

various methods for recording 'handshakes' can be compared

if 5 people shake hands with the 4 other people this suggests there would be 20 handshakes

why do you half this?

the 'handshake' numbers are triangular numbers
but whereas the rule for the number of handshakes for 'n' people is n(n - 1)/2
the rule for the nth triangular number is n(n + 1)/2

the nrich site has a demonstration of how two triangular numbers fit together

this work will hopefully lead to an appreciation of a general rule which can be developed to consider how many handshakes there would be for e.g. the class, year, whole school and the locality

you could then move on to consider properties of triangular numbers

which triangular numbers are particularly interesting?

what digits do triangular numbers not end with?
what happens when you add two consecutive triangular numbers together?

why is that?
which two sum to 2500?
which sum to 10,000?

which triangular numbers are multiples of 11?
why?

why is 1 + 2 + 3 + 4 a triangle?

how can odd 'n' triangular numbers be reformed into a rectangle? the even 'n' triangular numbers?

some of these questions are not easily answered ...

'mystic roses' link with handshakes
how?
the 'nrich' site has a very useful programme for drawing 'mystic' roses

stacks of cans give a visual form to triangular numbers

as well as counting cans in growing stacks you can analyse how many cans touch 1 other, 2 others, 3 others etc.

you could explore stacks of 'cans' in 3 dimensions:

how many oranges are there here?

how many sweets?
- assuming that there are layers of sweets (which I don't think is the case)

triangular numbers occur when you select 2 things from a number, 'n'
you can also extend the considerations by looking at different ways to choose e.g. 3 from 5 (working towards 3 from 'n')

bearing in mind that the selection e.g. blue, red, green is the same as a rearrangement, e.g. red, green, blue etc.