### 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.