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Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

### 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?

### Why do this problem?

This problem looks at the link between a situation described in words and the same situation described graphically. Learners interpret motion as shown on a distance time graph and then investigate the freedoms they have when drawing a graph based on given information. The freedom and constraints naturally provoke rich discussion when this task is approached in small groups.

### Possible approach

Sketch one of the graphs on the board. Ask for suggestions for what the two axes could represent and encourage discussion about the story the graph would be telling.
Introduce the idea that the $x$ axis shows the time in minutes since noon, and the $y$ axis shows the distance travelled. Show the three graphs of the car and the scooter. In pairs, learners could discuss what the graphs tell them about the motion of the two vehicles. It is worth discussing the idea that "overtaking" means one vehicle passing another travelling in the same direction, whereas "meeting" usually means that the two vehicles are travelling in opposite directions. Another area for discussion could be to suggest a scale and units for the $y$ axis, based on learners' knowledge of cars and scooters.

For the five statements, learners could each sketch what they think the graph could look like, and then compare their answers in pairs or small groups. This can lead to fruitful discussion about the ways in which the statements constrain the appearance of the graph and what freedom they have.

After working on the last part of the problem to produce a graph showing all the statements, learners could try On the Road which investigates the meeting time of the bike and the scooter.

### Key questions

What is the same on all three graphs? What is different?
What is fixed by the information given in the statements? What can be changed?
Is there only one possible graph for each statement?

### Possible extension

Learners could try On the Road straight away without having the example graphs first.

### Possible support

A worksheet with examples of graphs is available here. Learners can then identify which graph correctly represents each statement.