Why do this problem:
Averages can seem rather common-place, we think we know all about
them, so this problem takes 'average speed', a concept many
students at this Stage will think they know, and with a simple
question probes their understanding.
Possible approach :
The questions below, the support activity, and the suggested
extension task indicate a route into the concept of 'average
For abler students grasping that any rate has a reciprocal form and
that considering that alternative form might sometimes
be useful is an important insight. (Miles per hour, is
the reciprocal of hours per mile, for example)
Key questions :
What does 60mph mean ?
Why might someone think that the average of 50mph and 70 mph was
60 mph ?
Does it matter how far Cardiff is from Cambridge ?
What would change if it was twice as far ? Half as far ? 100
miles ? One mile ?
Possible extension :
- Can you write a formula that connects the two average speeds
(there and back) with the average speed for the journey ?
- What happens when one of the two average speeds (there or back)
is extremely large or extremely small ?
Possible support :
Students that do not already have a properly grounded understanding
of speed as a rate of change of distance (or displacement) over
time could benefit from some practical, tangible experience.
Take a marked distance along the floor or the wall for example and
'step' along it with 'finger footsteps' using a stopwatch to
determine the time for the journey. A 'finger footstep' is the
distance between the thumb and index finger when there is a wide
'V' between them, or from thumb to 'baby' finger like the
children's hand game 'incy wincy spider'. Students can be asked to
make leisurely journeys, or fastest possible, and each time work
out the average speed in 'footsteps' per minute. Encourage
questions and challenges as students begin to give meaning to these
measures and visualise the 'journey' when given the 'speed'
(footsteps per minute).