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Distance-time graph inquiry

This prompt was designed by Mark Greenaway (an advanced skills teacher in the UK) after a discussion about the nature of inquiry prompts. Mark contacted inquiry maths with an idea for a prompt. He proposed to play the first part of a YouTube clip of a drag race and encourage students to ask questions.
       
This idea seemed similar to Dan Meyer's 3-Act lessons with an initial stimulus set in a practical or real world context (see box below). In inquiry maths lessons, the structure of a lesson is not determined beforehand. Students are invited to play a large part in determining how the inquiry proceeds. It is also the case that many prompts are deliberately 'de-contextualised' to encourage students to provide their own context (and that context might indeed be a race).
Dan Meyer's 3-Act lesson
3-Act lessons follow a 'mathematical story':
Act One The teacher introduces the central conflict of a story "clearly, visually, viscerally, using as few words as possible."
Act Two The student overcomes obstacles, looks for resources, and develops new tools.
Act Three The teacher oversees the resolution of the conflict and sets up a sequel or extension problem.
      
In view of these considerations, Mark designed a first version of the diagram above without labels on the axes. This prompt could have led to a very open, multi-faceted inquiry with students interpreting the graph in different ways. However, as the topic was distance-time graphs, Mark gave students more structure by labelling the axes - thereby hinting at possible contexts for the inquiry.
         
The students' questions and comments about the prompt (below) show a high degree of knowledge about the graph, although still with questions and ambiguities to be resolved:
  
    
The responses suggest the class could  move quickly onto changing features of the prompt and exploring new situations. Perhaps, they could change one of the axes to 'speed', especially as it appears in the formula. Could they plot speed against time and speed against distance for the same journey. How do the three graphs compare?
   
Alternative prompts   
During the development of this inquiry, Mark suggested two other prompts. The attraction of the first one (right) lies in extending students who are comfortable with distance-time graphs. Students' questions about the two 'journeys' might be: How could we calculate who travelled the greatest distance? How does the graph link to acceleration? How would we calculate the acceleration for each part of the journey? How could we show acceleration on a time-distance graph?
  
The second prompt, in presenting two calculations, challenges students to compare and contrast. Why is the first statement correct when the second is not? What would happen if the journeys were not split into two equal amounts of time, or two equal distances?


Mark Greenaway runs his own website, which is highly recommended for its comprehensive coverage of ideas for the mathematics classroom. You can follow Mark on twitter @suffolkmaths.

Resources 
Promethean flipchart    download

Inquiring into contexts
These are the questions and comments of a year 10 class at Magdalen College School (Cambridge, UK). The students used the regulatory cards to direct the inquiry towards devising contexts for the graph. Luke Pearce, the class teacher, reports on the inquiry in a post on his blog and you can view slides that show the course of the inquiry here.

You can follow Luke on twitter @discoverymaths.
  
Misconceptions about the prompt   
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 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.

Nichola Sowinska
is a maths teacher in Peterborough (UK). She is studying for a Masters degree at the Institute of Education, London. You can follow Nichola on twitter @NixxSunshine.
  
The prompt in a secondary classroom
James Thorpe (a UK maths teacher) used the distance-time graph prompt with his year 10 class (set 4). 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 into the next lesson, James steered students towards the more challenging activity of 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 photo shows one student's interpretation of the graph after the initial discussion.

James Thorpe was a maths teacher at John Taylor High School, Staffordshire (UK) at the time of the inquiry. He taught parts of the maths curriculum through inquiry and devised his own prompts.
    
Open inquiry in year 7
Nicola Stokes (Head of Mathematics at Seahaven Academy, UK) sent the questions and comments of her year 7 class (left) to the website. We note the initials next to the contributions, which allow a teacher to assess students' mathematical sophistication when reviewing the inquiry. Interestingly, the students identify what is missing from the graph and discuss its shape and type. They do not, however, speculate about what the graph represents. As it was the students' first inquiry, we might view this as a reluctance to take a risk. Alternatively, it might indicate that, as a first experience of inquiry, the students found such an open prompt difficult to interpret without, for example, the axes labelled.