median
don steward
mathematics teaching 10 ~ 16

Saturday, 17 March 2012

Diffy

an intriguing task
with lots of subtracting to be done
and the tasks provide a sensible reason for introducing algebra (unlike life in general...)

start with any four, smallish, numbers - in any order, initially
calculate the positive difference between an adjacent pair of numbers, looping back to the start for the right-hand end number, to produce a new set of 4 numbers
keep on doing this, line by line
until you have a good reason to stop

Herbert Wills analysed this task in 1971 and gave it the name Diffy

the work was brought to my attention by a fine booklet from 'Motivated Math Project' by Stanley Bezuska at Boston College Mathematics Institute (published in 1976, I think)

you can return to the basic task of doing a Diffy, year on year, with fresh and increasingly complicated starting four numbers - chosen as consecutive terms from various 'standard' number patterns (see below)

various algebraic skills can be practiced for ever more complicated number patterns to establish a generalisation for the number of steps it always seems to take to reach 0 0 0 0 

all starting arrangements of four numbers, reduce to 0 0 0 0 quite quickly in most cases - usually in fewer than 7 steps
it's easy to make errors and tedious to check, so it can be helpful to set up a spreadsheet in advance
(using abs(difference between cells))

to begin the task(s):
ask students for any 4 numbers (not too big and not in any order) and then go through the 'diffy' process, without explanation - they try to sort out what the rules are...


it is quite hard to find a set of numbers that involves more than six steps (iterations)
but it is possible e.g. :




after a while playing around with any four numbers trying to better the "class (world) record" diffy

start to input four consecutive terms of a sequence and explore what happens
e.g. for a constant difference pattern:




















following this sequence of steps:
  • try out several particular examples
  • see what patterns are common to all the examples (or a few at least)
  • decide how many steps a 'diffy' seems to take for a particular number pattern
  • prove this using algebra
at various stages (maybe years) , the work can involve:
  • consecutive multiples (start with a number keep multiplying by e.g. 2)
  • a linear rule: start with a number, multiply by e.g. 3 and e.g. subtract 2 each time
  • consecutive fibonacci numbers
  • consecutive square numbers
  • consecutive triangular numbers
  • consecutive cubes
  • consecutive terms of a general geometric sequence

these are all included on a powerpoint

Puntmat have an interesting variation of this task, using the NLVM interactive square, asking students to find a sequence of particular numbers after four iterations (steps) 



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