This
problem is short and encourages students to think about the
meaning of place value and engages their logical thinking. It could
be used as a starter to engage pupils as they come in to the
classroom, though there are good extensions available for a full
lesson.
Possible approach
Put the problem on the board to allow pupils to familiarise
themselves with the problem.
Discuss as a group the possible forms of proper 6 digits
numbers. Students might like to decide whether numbers starting
with zero count as a proper six-digit number (no!).
Allow students to experiment to try to determine the number of
possible answers.
A good problem solving strategy is to make the problem
smaller, e.g.how many three- (or four-) digit numbers do not
contain a $5$, then to work out how to extend the solution
method.
Key questions
Is $000001$ a six-digit number?
How many six-digit numbers are there?
How many choices do we have for the first digit?
How many choices do we have for the second digit?
Possible extension
How many six-digit numbers do not contain a $5$ or a $7$?
How many six-digit numbers are there for which the digits
increase from left to right (such as $134689$ or $356789$)?
How many numbers less than $10$ million do not contain a
$5$?
Will your methods extend to similar problems? if so, can you
express them algebraically?
What other [interesting] questions could you ask starting "How
many six-digit numbers..."?
Possible support
You could ask the almost equivalent question "How many
six-figure telephone numbers do not contain a $5$?". This
encourages student to imagine dialling a number in sequence, which
will may help them to see the different choices which can be made
at each step of the process.
Encourage students to adapt the problem to make it accessible:
fewer digits, how many six-digit numbers are a multiple of $10$
(probably seen as a number with $0$ as last digit) or even, or a
mult of $5$, or square, etc.