Graphing stories
What is a graphing story?
A graphing story is a classroom routine that can be used to develop secondary school students' knowledge of graphs, relationships and mathematical models. It was pioneered by the American math teacher Dan Meyer. Students see short films of everyday events, and then use graphs to describe them. In the ensuing discussion, there is a natural need to introduce mathematical concepts such as linear relationship, proportionality, exponential function and derivatives.
The idea is simple: Students are shown a short film of an everyday event, such as a glass filling with water, a piece of salmon cooking in the oven, or a swing swinging back and forth. The students are then prompted to sketch a graph that describes the event, e.g. how they think the height of the water, the temperature of the salmon or the speed of the swing changes over time. The students' sketches become the starting point for a classroom discussion, which ends with the correct graph being displayed. Graph paper to hand out to students can be downloaded here.
The graphing story videos here at www.matemagi.com usually follow the same pattern: First, an event is shown. Then a coordinate system is displayed that reveals what the students are expected to graph. Is it how the height of the water in the glass changes with time? Or maybe how the volume changes with the height of the water? Then the students get to see the event again, so that they can sketch their graph. If necessary, you can rewind and show the event several times. Finally, the correct graph is displayed. Below you see an example.
Workflow
Show the first part of the film to the students. Pause after the coordinate system has been displayed and explain to the students that they are going to draw a graph that describes the event. Tell them that they don’t have all the information needed to draw the graph. That means that they will have to make some assumptions and estimates. In the graphing story above, for instance, the students need to estimate the height of the water.
Distribute graph paper or have students draw a coordinate system in their notebooks.
Start the film again, so that the students can see the event once more, and give the students time to draw the graph.
Walk around the classroom and follow the students' work. Select some graphs, that you want to compare in the classroom discussion.
Discuss some of the students' graphs. This can be done by letting students describe their graphs orally, while you draw them on the board, or by letting the students draw their graphs on the board. Document camera and projector, or a digital submission system, are other possibilities. Compare students 'graphs and help them formulate the graphs' similarities and differences. At the same time, introduce important mathematical concepts.
Give students time to revise their graph based on what they’ve heard in the discussion.
Restart the video and display the correct graph. Discuss any similarities and differences between the correct graph and the students' suggestions.
First time
The first time you work with a graphing story, it’s a good idea to model for the students how to go about it. A good approach is to play a film that shows an event and then "think aloud" while sketching the graph, for example by showing how to answer the following questions:
At what point should the graph begin?
“From the beginning, there is no water in the glass. The height of the water is thus 0 cm, at the time 0 seconds. This means that the graph starts at the origin.”
At what point should the graph end?
“At the end, the height of the water is 10 cm. At that point, 10 seconds have passed. Thus the graph ends at the point (10, 10)."
What do I think the graph looks like in between?
“The water is poured at a steady speed into a cylindrical glass. Therefore, the height also increases at a steady speed. That means the graph is a straight line.”
The next step is to play a similar film and let the students discuss the same questions in pairs. Then each pair sketches a graph. When students are more accustomed to graphing stories, they can sketch their graph individually and then compare their graph with a peer.
Variation
There are lots of ways to vary the work with graphing stories. For example, instead of drawing the graph that describes the event, students can choose between four graphs and justify their choice. That approach can make common misconceptions visible. In the description of most of the graphing stories, there are suggestions for such graphs.
You can also follow up the graphing stories with questions. In the example of water being poured into a glass, some such questions could be:
What is the equation of the line?
How does the graph change if…
the water is poured faster
the water is paused for a few seconds
the glass is higher
the glass has a larger diameter
It is also possible to work with one and the same graphing story in different year groups. In lower secondary school, for example, students can describe how the speed of a swing changes over time when it is in motion. In upper secondary school, in contrast, students can be given the task of finding the expression of the trigonometric function that describes the speed as a function of time.
The work with a graphing story can have different focuses. Sometimes you can be interested in the students sketching the graph, without caring about any grading on the axis. Other times, you may choose to distribute coordinate systems with a suitable scale on the axis or let the students set out a suitable scale themselves.
Why use graphing stories?
For me, there are several reasons to work with graphing stories.
By working with graphing stories, students experience how graphs are used to describe everyday phenomena. It connects mathematics to students' reality and allows them to see the usefulness of mathematics.
The classroom discussion of the students' graphs, creates a need to formulate what the students have drawn. That provides an opportunity to introduce important concepts, such as slope, linear, constant, growing, decreasing, etc. In upper secondary school, you can use graphing stories to discuss more advanced concepts, such as derivatives, inflection point and maximum.
After working with several different graphing stories, a natural step is to compare the graphs and categorize them. This makes graphing stories an excellent tool for introducing and naming different types of relationships, e.g. linear, quadratic, periodic and exponential.
In lower secondary school in Sweden, it is common to work mainly with linear relationships. With the help of graphing stories, you can let students experience that there are other types of relationships, whose graphs are not straight lines.
To sketch graphs of everyday events, is a common task in many textbooks. Letting students see a film of the event, makes it more concrete, which can make it easier for students to draw the graph. In addition, the connection between event and graph becomes stronger.