James Thorpe, a teacher of mathematics at John Taylor High School, Staffordshire (UK), used the distance-time graph prompt with his year 10 class. He started the inquiry by showing the graph without the distance and time labels to see what the class thought it could be about. James reports that the class made insightful observations about the prompt, coming up with good ideas that included stock values and temperature. On being directed towards distance and time, the class got into a debate about what the negative displacement meant, resolving the issue by, as James says, "some good acting out of the graph.” 

The inquiry developed onto the students drawing their own graphs, which they also acted out. James concludes that “this showed their understanding of how the graph represents motion, direction of travel, periods of rest and how the gradient shows the speed. All in all an enjoyable lesson!”

In planning for the second lesson, the students wanted to consolidate their understanding, so James provided a task in which they had to match distance-time graphs with different contexts. They also began to calculate speeds. As the inquiry continued, James steered students towards graphing a ‘journey’ on mini-whiteboards as it was acted out by a pair of their peers. 

James was pleased with the results: “I was impressed with how well students did in drawing the graphs. Every pair got to act their graph out and I would say in most cases students got close to drawing an accurate graph for their journey.” James concludes by saying, “Thanks for the inspiration - it is definitely a prompt that can inspire a lot of thought. It is interesting that the line under the horizontal axis actually proved to be useful." 

The prompt often uncovers misconceptions students hold about graphs. Many of these originate in the inability to interpret the graph as a mathematical relationship between two variables. Rather, students view the graph as a literal representation of something within their everyday experience. 

For example, when Nichola Sowinska, a teacher of mathematics in Peterborough (UK), used the prompt, a student described the graph as showing a man climbing a mountain and descending on the other side. Another student asked, “Do you have to walk backwards to go the other way?” 

Such misconceptions can be tackled by the teacher asking what labels to put on the axes. 

Generally, classes suggest ‘time’ for the horizontal axis. When the vertical axis is labelled ‘distance from the starting point’, students can (be guided to) realise their error in interpreting the graph as a pictorial representation of climbing a mountain.

However, other interpretations that seem to view the graph as a picture might be justifiable when the label on the vertical axis is different. Another suggestion from Nichola’s class involved a bird flying into the air, hovering, and then diving into the sea to catch a fish. If the vertical axis is ‘distance above sea level’, then the graph shows the flight as described, although the teacher should check that the student understands that there is no implication of horizontal displacement.