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Why do this problem?
This
challenge requires a logical, systematic approach. The solution
might be unexpected to some, and might cause disagreement amonst
students. Whilst the problem might seem to be algebraic, it is more
an exercise in visualisation, time and motion.
Possible approach
Ask students to tell their neighbours their initial guesses as
to the number of buses that will be passed. Give students time to
work on paper, checking out their ideas and developing convincing
arguments. Ask for volunteers to explain their ideas to the group.
There is likely to be disagreement which (ideally) should prompt
clarity of explanation.
It could be useful to model the process with the students as
follows:
Split some volunteers into two queues of buses, one on each side of
the room.
Set up three intermediate bus stops. Every ten seconds ask each bus
to move to the next stop.
They must count each time they walk past someone. Continue until at
least 6 buses have crossed in each direction.
How many crossings did each person make?
Is there anything about the journey that is more obvious now that
it has been acted out?
Students could be asked to write a full clear solution. Some
of these could be shared with the group.
How convincing and clear did the group find these
explanations?
This may be an opportunity to introduce a distance-time graph
to represent the situation. It demonstrates the number of solutions
very elegantly, but students often find these graphs confusing. The
teacher could put a completed graph on the board, and ask the
students to make sense of it. It would be useful to discuss the
meaning of:
a
horizontal strip - the view from a particular point on the
route,
a
vertical strip - a summary of all bus positions at a
specific moment in time,
a
diagonal strip - a description of the journey for a
particular bus.
Emphasise that this is not an artistic picture of events, and
certainly not a road map, but it summarises a lot of information,
to give an overview of a complicated situation.
Key questions
- Is there a difference between the first bus of the day and a
bus which sets off later on in the day?
- Does each bus make the same number of crossings?
- Is there a clear way of recording the motion of the buses?
Possible extension
Investigate the number of crossings if the journey time and/or
gap between buses is altered.
Does it matter if the gap time isn't a factor of the journey
time?
Let the bus journey be up (or down) a long hill - so that the
speed in one direction is different to the speed on the return
journey.
Can students summarise/generalise their solutions to the
variations they have tried?
Possible support
Simplify the numbers - eg 30 minute journey time.
Provide counters to allow students to model the motion of the
buses, establishing bus stops at 10 minute intervals to make the
movement easier.