### On the Road

Four vehicles travelled on a road. What can you deduce from the times that they met?

### There and Back

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

### Escalator

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

# An Average Average Speed

### 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 speed'.

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).