Why do this problem?
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.