### Why do this problem?

This
problem offers an interesting and challenging exercise in place
value. It can be solved using an experimental approach or more
formally using algebra. The task offers rich extension
possibilities exploring a context for 'clock arithmetic'.

### Possible approach

Ask students to work in pairs to decide what the two meters
will read after 1 mile, 10 miles, and 100 miles, and then give them
time to consider the first part of the problem. There is likely to
be some confusion because of the presence of 1/10ths of a mile on
just one of the meters. Some students could be asked to feedback on
what was tricky/hard to explain/ hard to agree on.

In order to establish a good approach to these questions, aska
few different pairs to demonstrate their first ideas/full reasoning
at the board. Then ask all students to derive the answer (4953)
writing it out clearly, before making up their own initial meter
readings and calculating how far they must go before the meters
match.

On the board putthe headings "these pairs of meters will never
match" and "these will match". Students can record pairs of initial
values for their peers to check. At some stage you'll probably need
to declare the no-matches list, closed. Students who feel stuck
could take a little break to test these 'solutions' and to try to
observe what is making them work.

Some students might like to try to develop an algebraic
expression for the distances. How can the question be rephrased as
an equation?

### Key questions

- What is the shortest trip that will cause the milometer to
change? What effect will this have on the trip meter?
- How can we get the last digit to match?

### Possible extension

Once students have found pairs which work, they might like to
explore the richer questions involved when the milometer goes round
the clock.

- What happens when the car has travelled more than 10,000 miles?
Does this allow any more possible starting numbers?
- What journeys leave the digits on both clocks unchanged?

### Possible support

You could start students off with a simpler question, e.g. if
the trip meter registered 000.0 miles and the milometer registered
00009 miles or 00234 miles, how many miles would the car have to
travel for the digits to work? When working on the starting
question (4631 etc) allow a lot of time for trial and error
solutions, encouraging paired discussions on how to make a better
trial each time.

Encourage students to lay out the readings from the two dials
in place-value columns and to work one step at a time, recording
each new reading in turn.