don steward
mathematics teaching 10 ~ 16

Sunday, 1 March 2015

division cycling

a task that might possibly make long division a bit more interesting
(a long divided by a short anyway)

the main task:

 what do students notice?

dividing by 4 is the simplest place to start
can students create other numbers so that when you divide by 4 they 'cycle'
[i.e. the lead digit goes to the end]

you can either work backwards or forwards to create these numbers

with division by 4, all of the digits will work as a lead number
there are some patterns to notice, families: starting with 2, 5 and 8 for example

dividing by other numbers is also interesting:

all divisors work (create a cycle)
unfortunately the lengths of the numbers for other divisors are rather long:
  • dividing by 2 needs a number that is 18 digits long
  • 3 needs 28
  • 4 all need 6
  • 5 needs 42 apart from the one question (starting with a 7) above
  • 6 needs 58 (not really for the faint hearted)
  • 7 needs 22
  • 8 all need 13
  • 9 needs 44

however, this work can be ever so good tables practic

it's interesting, if peculiar
that there is only one six long example for dividing by 5 and the division yields the repeating block for 1/7th as a decimal...

patterns when dividing by 4:
if you chop up the six digit numbers into two blocks of 3 
and add them e.g. 205 + 128 you get interesting results

as you do if you chop them into three blocks of 2 and add them

all reminiscent of turning fractions into decimals with prime divisors

Ed Southall has kindly posted some slides for this on his blog

and here's the T shirt:

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