# Four on the Road

Four vehicles travel along a road one afternoon. Can you make sense
of the graphs showing their motion?

## Problem

Caroline sets out in her car on a journey. Scott is out on his scooter that afternoon too. Each of these graphs represents a journey they could make. The horizontal axis shows the number of minutes that have passed since noon, and the vertical axis shows the distance each vehicle is from Caroline's house.

Image

Can you tell a story for each of the journeys shown in the graphs?

What is the same about each journey? What is different?

You may wish to use words like speed, direction, distance, time, overtake, and meet.

Caroline and Scott also see Muna on her motorbike and Bill on his bike that afternoon.

Here are five statements about all their journeys.

- Caroline overtook Scott at 12:30pm
- Caroline met Bill at 1pm
- Caroline met Muna at 1:30pm
- Muna met Scott at 1:45pm
- Muna overtook Bill at 2pm

Sketch graphs which represent each statement. Are there other graphs you could have drawn? How much freedom do you have? What extra statements could you add to reduce this freedom and specify the graphs more exactly?

Can you draw a graph showing all four vehicles which satisfies all five statements at the same time?

## Getting Started

What do the two graphs look like if one vehicle overtakes the
other?

What do the two graphs look like if one vehicle meets the other?

Think about the gradients and crossing points for each case.

What do the two graphs look like if one vehicle meets the other?

Think about the gradients and crossing points for each case.

## Student Solutions

Well done to Thomas from Woodland Middle School, and Rayan from St James School, who each sent us graphs which show all the pieces of information.

Image

Image

These aren't the only graphs which solve the problem, so why not try drawing some more and looking at the similarities and differences between them?

## Teachers' Resources

### 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?