powerpoint is here

division practice

I think I've seen these called 'hiccup' numbers somewhere

a powerpoint is here






maybe there is someone out there who can enjoy dividing by 73

the powerpoint is here
see also practice makes perfect blog division

a ppt is here

connected results, opportunities for spin-off

maybe slightly too many questions....
(not using a calculator, especially a scientific one, unless it's to check results)



the equivalent fractions below
involve all of the digits, 1 to 9

there are several more 1 to 9 versions for some of the fractions

reference: ben vitalis at fun with num3ers
many thanks to him

practice at multiplying and dividing


there is another one for a third

a powerpoint is here

shows some of the divisions
and some of the features
including links to the recurring decimal form of 1/7th

plenty of practice with the 7 times table

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:

choose two digits (e.g. 2 and 5)

• divide the sum of their squares by their sum (i.e. (4 + 25) divided by 7 for the example)
• write this as a number and a remainder: (i.e. 29 / 7 = 4 rem 1)

try a few of these

something special happens for two consecutive numbers
what is it?

a proof of this property involves (for n and (n + 1)):
splitting the numerator into 2n^2 + n + (n + 1)
dividing by (2n + 1)
that gives n  +  (n + 1)/(2n + 1)
which is  n  rem  (n + 1)

divide 820512 by 4
what happens?

what happens if you divide 615384 by 4?

























so, using this property for special numbers when dividing by 4,
can 6 digit special numbers be found that
start with the other digits?

[yes they can]

you could start these tasks by exploring what happens to a 2-digit number when the front digit is put to the back and the two numbers are (a) added and (b) subtracted






















and what happens with a 2-digit number if you square them first and then subtract?













establish that the result will always divide by 9 and by 11

every student ought to know how to divide by 5 in their heads




they might also appreciate the reasons why a rule works

a five digit key pad is set to a number such that the number with a 1 after it is 3 times larger as it is with a 1 before it

what is the code number?

[Moscow puzzles 253]

the numbers below are particularly great because
(i) they each involve all of the digits (0 to 9) and
(b) they divide by all the numbers from 1 up to 18

so students can be set tasks involving a long number divided by a much shorter one - of your or their choice


what is the hcf of these numbers?