Graphs represent information. Much information nowadays is presented in graphs and they are used by all kinds of people, in all sorts of contexts. Knowing how to interpret graphs is important to make sense of the world around us and it has also become a crucial skill to have in the workplace.
People can find reading and interpreting graphs difficult because they can be confusing and misleading. This unit aims to help you to support your students in acquiring the skills of interpreting and constructing graphs to prepare them better for life. It does not focus on the basics of graphs, such as constructing coordinates, plotting points precisely or using equations and formulae to find out the gradient or points of inflection. Instead, the unit aims to help you make working with graphs more accessible to your students by developing a narrative or story to assist them with their understanding of graphs. Underpinning this approach is the notion that ‘every graph tells a story’.
Copies of the graphs and other resources can be found in Resources 1 and 2.
The learning in this unit links to the NCFTE (2005, 2009) teaching requirements specified in Resource 3.
Research shows that asking students to develop a story or narrative as part of their learning activities can help their understanding. Bruner (1986) argues that this is the case because ‘human beings are essentially narrative animals, telling stories to themselves and others as a way of making sense of the world’ (Mason and JohnsonWilder, 2004, p. 68). This unit will help students to develop such storytelling in the context of making sense of graphs. The underlying theme across all the activities is the notion that ‘every graph tells a story’.
Activity 1 focuses on finding out what makes interpreting graphs so hard. In previous years, you may have told your students the factors that are important to include in a graph. Here, the students themselves work together to generate ideas about this.
Before attempting to use the activities in this unit with your students, it would be a good idea to complete all (or at least part) of the activities yourself. It would be even better if you could try them out with a colleague, as that will help you when you reflect on the experience. Trying the activities yourself will mean that you get insights into learners’ experiences that can in turn influence your teaching and your experiences as a teacher. When you are ready, use the activities with your students. After the lesson, think about the way that the activity went and the learning that happened. This will help you to develop a more learnerfocused teaching environment.
This activity works well if done in groups of four students, because more examples are then available for them to examine. Ask your students to collect and bring to school examples of graphs that they have seen from different contexts: newspapers, adverts, brochures, etc.
Explain to your students what they are required to do in this activity using the following prompts:
As a class, develop a list of ‘good things to do when constructing graphs’ from the groups’ ideas. Pin this up on the wall so that students can refer to it during later graph work.
This is the account of a teacher who tried Activity 1 with his secondary students.
This activity did not get off to a good start. I had asked students to bring in examples of graphs and they managed to bring in … none. Perhaps it was lack of motivation, or perhaps it was that they did not know where to find any graphs. To motivate them I asked them to think of examples where graphs are used in real life or in the workplace, and where there could be problems if the graphs are not interpreted correctly.
To give them an example, I showed them the magazine and newspaper that I had brought in with some examples of graphs. By the next lesson most students had brought in several examples. Some even managed to download some graphs from the internet.
The students worked in groups of four. They found the graphs related to games and advertisements easy and decided this was because the information it represented was simple. They did notice, however, that some of these ‘easy’ graphs used scales that were not clear, and the labels in the axes were also not always correct. They had more problems with graphs representing medical and economic information: they found it hard identifying the variables and interpreting the relations between the variables.
I collected the graphs to use for later activities and told the students to bring in more so we could have a whole library of graphs to use for years to come.
When you do such an activity with your class afterwards, reflect on what went well and what went less well. Consider the questions that led to the students being interested and being able to get on and those where you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting, as Mr Chadha did, some quite small things that made a difference.
Pause for thought Good questions to trigger such reflection are:

What are the variables (or characters) in the story that graphs tell?
The first thing students have to understand in order to read graphs is that they are a visual representation of the relationship between variables. This is important and yet it is often overlooked. Information regarding which variables these are can be found by looking at the labels on the axes. For example, a distance/time graph shows the relationship between time lapsed (or the time of day) and the distance travelled. The xaxis (horizontal) tends to be labelled with the time variable, the yaxis (vertical) with distance.
The first part of Activity 2 is a card sort to make students aware of the variables involved in the graphs. This type of activity helps you to evaluate student learning. You need to keep the group size small to enable all students to engage in the learning activity. In Part 2, the students are asked to use similar approaches with the examples they looked at in Activity 1. The card sort consists of two types of cards: some with graphs and some with descriptions. The activity involves matching a description to a graph and being able to explain the reasons for that choice.
Prepare enough copies of the card sort in Resource 1.
This task works well for work in pairs or in threes. It is less effective for bigger groups because the students will not all be able to see and easily read what is on the cards.
Remember to tell the class that ‘every graph tells a story’.
Ask the students to look at the card and match the descriptor to the graph.
Tell them that it can be helpful to use the following phrases:
Explain to your students what they are required to do in this activity, using the following prompts:
Read Resource 4, ‘Using pair work’, to find out more.
Getting organised for the card sort meant that I had to get several copies of the cards and cut them out – which is timeconsuming and also costs money. However, I have since used them with other classes and colleagues have borrowed them as well, so it has been worth it.
I first asked the students to work in pairs on the card sort and to think carefully about the justification for their choices. I then asked them to compare their answers with the pairs of students sitting in front or behind them. What I really liked about these card sorts is that they asked students to focus on variables. They also triggered some really good mathematical discussions.
The students found making their own card sorts straightforward for the graphs in the ‘easier’ pile, but more difficult for those in the ‘harder’ pile – as I had expected. To help them with the harder graphs, I asked them: ‘What was it you did with the easy graphs?’ This helped them to get ‘unstuck’. As a result of this activity, many of the graphs in the harder pile moved to the easier pile. I also asked them to amend the notes they made for the last part of Activity 1, where they created a list of ‘good things to do when constructing graphs’.
By asking the students to bring in examples of graphs, working on developing our own tasks and exchanging these, we now have an impressive amount of teaching material about a large variety of graphs in the class! The students feel proud and seem more selfassured, because they actually made the teaching resources themselves.
Pause for thought

The previous activity asked students to identify the variables, or characters, in the ‘story’ that a graph represents. Now it is time to find out what happens between these ‘characters’. In mathematical terms, this means examining the graph for how the variables relate to each other at different points and whether that relationship changes.
Students often find it difficult to interpret what a gradient signifies, especially in distance/time graphs. Activity 3 asks the students to think about this through a card sort. Activity 4 then asks the students to use their knowledge about reading and interpreting graphs by making up a story as reporters for the Big Rickshaw Race.
Prepare enough copies of the card sort in Resource 2.
This activity works well in pairs or in threes. It is less effective for bigger groups, because the students will not all be able to see and read what is on the cards easily.
Remember to tell the class that ‘every graph tells a story’.
Explain to your students what they are required to do in this activity, using the following prompts:
Explain to your students what they are required to do in this activity, using the prompts below.
Figure 2 is a graph that represents the motion of two of the auto rickshaws racing each other as they approach and go around a bend in the road in the annual Big Auto Rickshaw Race.
Try to visualise what is going on and answer the following questions:
I thought that using Activity 3 would be too easy for the students. However, while we were using it, both I and the students became aware of the many misconceptions they had. I asked them to work on the card sort in pairs at first. We then had wholeclass discussions that I orchestrated by holding up a graph card and asking what the story was, or holding up a descriptor card and asking what the matching graph was. I insisted on the students giving their justifications for what they said. After each justification, I asked the class who agreed, who disagreed and who was not too sure about the justification. This is how the misconceptions got aired and worked on. The task did take longer than I had anticipated but it was very worthwhile.
Activity 3 gave the students the knowledge and confidence to tackle Activity 4. The graph seemed simple at first, but the questions drew attention to some unusual elements – the students expressed surprise when they noticed them. The part where they had to be a reporter was funny. At first they were rather shy about engaging with it, so I invited Shamira, who I know loves acting, to come and sit at my desk and pretend to be the reporter. She had a go and I then asked others to comment constructively on what she had said, including on the interpretation of the graph. Shamira adjusted her story and had another go at it. After that the students were happy to try it out for themselves. Some did it in pairs, some in threes or fours. What I liked was that, while they were ‘reporting’, they were looking at the graph and interpreting the information that was there.
There was an addition to this activity that I loved. During break time I saw two students ‘racing’ against each other and another student pretending to be reporting on the race, including mimicking holding a microphone.
Pause for thought

So far in this unit, the focus has been on finding out what story a graph tells; that is, the interpretation of graphs. Constructing and interpreting graphs are very closely related; however, constructing graphs also involves another way of thinking. Instead of simply inventing a plausible story, it also requires representing the story in a graph and paying attention to all its elements, then checking whether the graph actually tells the story that was intended.
To practise these skills, the next activity asks the students to write a film script for an action movie and to construct a graph to tell that story. You might ask your students to spend five minutes at the start choosing their favourite action hero  they might choose Shaktimaan or Kangna Ranaut from Krrish or even James Bond.
Explain to your students what they are required to do in this activity, using the following prompts:
You are writing a script for a big action movie! The scene you are working on requires the hero to get away as far as they can from the place where they are in one hour – which is the mathematics classroom you are in now! It is a race against time. Any mode of transport can be used, including wheelbarrows or roller skates, as long as it is sensible and can be filmed. For example, a plane taking off from the school playground would not be sensible because planes require a long runway.
Describe your route and tell your story by drawing a graph.
You can read more in Resource 4, ‘Storytelling, songs, role play and drama’.
I was slightly apprehensive about doing this exercise. However, it sounded like a lot of fun and excitement, and an opportunity for the students to use their imagination. I should not have worried – they loved it! I did put some time limits on parts of the activity: 20 minutes to produce their script and graph, then another 20 minutes to swap stories and graphs and make changes. I used the stopwatch on my mobile phone to time this. Although I thought this might not be enough time, the students had no issues with it. It seemed to be part of the ‘race against time’ in the activity.
As a result, the class was energised and the students worked frantically. Some students got stuck and asked for help. I decided the best help they could get was from seeing examples, so I told them to ask their classmates whether they could have a look at their work. The effect of this seemed to be that those students worked in a supportive and collaborative way while still making their own stories, which was lovely to see.
The students ended up exchanging their graphs and stories with many other students. They were all curious to see what the others had done. They were very proud of their work and asked voluntarily whether they could take it home to show their family. I agreed. I also asked them to make an improved version of their work, with ‘perfect’ graphs that used a sensible scale, properly labelled axes and a title to then put on the walls in the classroom. I was very impressed by the quality of the mathematics in their stories and graphs.
Pause for thought

This unit explored ways of helping students make sense of graphs while becoming familiar with the ideas behind variables. From the start, the students were asked to relate their learning to the real world, bringing in graphs that they found in newspapers, adverts and anywhere else. The underpinning concept throughout was that ‘every graph tells a story’; this idea was reinforced by asking students to first match stories with graphs and then write their own stories for the graphs that they found. Modelling an action hero escape story using graphs provided an engaging and imaginative way to encourage the students to draw their own graphs, linking the variables appropriately so that the story emerged from their graph.
This resource is a card sort that you will find useful for Activity 2.
This resource is another card sort that you will find useful for Activity 4.
This unit links to the following teaching requirements of the NCF (2005) and NCFTE (2009), and will help you to meet those requirements:
In everyday situations people work alongside, speak and listen to others, and see what they do and how they do it. This is how people learn. As we talk to others, we discover new ideas and information. In classrooms, if everything is centred on the teacher, then most students do not get enough time to try out or demonstrate their learning or to ask questions. Some students may only give short answers and some may say nothing at all. In large classes, the situation is even worse, with only a small proportion of students saying anything at all.
Pair work is a natural way for students to talk and learn more. It gives them the chance to think and try out ideas and new language. It can provide a comfortable way for students to work through new skills and concepts, and works well in large classes.
Pair work is suitable for all ages and subjects. It is especially useful in multilingual, multigrade classes, because pairs can be arranged to help each other. It works best when you plan specific tasks and establish routines to manage pairs to make sure that all of your students are included, learning and progressing. Once these routines are established, you will find that students quickly get used to working in pairs and enjoy learning this way.
You can use a variety of pair work tasks depending on the intended outcome of the learning. The pair work task must be clear and appropriate so that working together helps learning more than working alone. By talking about their ideas, your students will automatically be thinking about and developing them further.
Pair work tasks could include:
Pair work is about involving all. Since students are different, pairs must be managed so that everyone knows what they have to do, what they are learning and what your expectations are. To establish pair work routines in your classroom, you should do the following:
During pair work, tell students how much time they have for each task and give regular time checks. Praise pairs who help each other and stay on task. Give pairs time to settle and find their own solutions – it can be tempting to get involved too quickly before students have had time to think and show what they can do. Most students enjoy the atmosphere of everyone talking and working. As you move around the class observing and listening, make notes of who is comfortable together, be alert to anyone who is not included, and note any common errors, good ideas or summary points.
At the end of the task you have a role in making connections between what the students have developed. You may select some pairs to show their work, or you may summarise this for them. Students like to feel a sense of achievement when working together. You don’t need to get every pair to report back – that would take too much time – but select students who you know from your observations will be able to make a positive contribution that will help others to learn. This might be an opportunity for students who are usually timid about contributing to build their confidence.
If you have given students a problem to solve, you could give a model answer and then ask them to discuss in pairs how to improve their answer. This will help them to think about their own learning and to learn from their mistakes.
If you are new to pair work, it is important to make notes on any changes you want to make to the task, timing or combinations of pairs. This is important because this is how you will learn and how you will improve your teaching. Organising successful pair work is linked to clear instructions and good time management, as well as succinct summarising – this all takes practice.
Students learn best when they are actively engaged in the learning experience. Your students can deepen their understanding of a topic by interacting with others and sharing their ideas. Storytelling, songs, role play and drama are some of the methods that can be used across a range of curriculum areas, including maths and science.
Stories help us make sense of our lives. Many traditional stories have been passed down from generation to generation. They were told to us when we were young and explain some of the rules and values of the society that we were born into.
Stories are a very powerful medium in the classroom: they can:
When you tell stories, be sure to make eye contact with students. They will enjoy it if you use different voices for different characters and vary the volume and tone of your voice by whispering or shouting at appropriate times, for example. Practise the key events of the story so that you can tell it orally, without a book, in your own words. You can bring in props such as objects or clothes to bring the story to life in the classroom. When you introduce a story, be sure to explain its purpose and alert students to what they might learn. You may need to introduce key vocabulary or alert them to the concepts that underpin the story. You may also consider bringing a traditional storyteller into school, but remember to ensure that what is to be learnt is clear to both the storyteller and the students.
Storytelling can prompt a number of student activities beyond listening. Students can be asked to note down all the colours mentioned in the story, draw pictures, recall key events, generate dialogue or change the ending. They can be divided into groups and given pictures or props to retell the story from another perspective. By analysing a story, students can be asked to identify fact from fiction, debate scientific explanations for phenomena or solve mathematical problems.
Asking the students to devise their own stories is a very powerful tool. If you give them structure, content and language to work within, the students can tell their own stories, even about quite difficult ideas in maths and science. In effect they are playing with ideas, exploring meaning and making the abstract understandable through the metaphor of their stories.
The use of songs and music in the classroom may allow different students to contribute, succeed and excel. Singing together has a bonding effect and can help to make all students feel included because individual performance is not in focus. The rhyme and rhythm in songs makes them easy to remember and helps language and speech development.
You may not be a confident singer yourself, but you are sure to have good singers in the class that you can call on to help you. You can use movement and gestures to enliven the song and help to convey meaning. You can use songs you know and change the words to fit your purpose. Songs are also a useful way to memorise and retain information – even formulas and lists can be put into a song or poem format. Your students might be quite inventive at generating songs or chants for revision purposes.
Role play is when students have a role to play and, during a small scenario, they speak and act in that role, adopting the behaviours and motives of the character they are playing. No script is provided but it is important that students are given enough information by the teacher to be able to assume the role. The students enacting the roles should also be encouraged to express their thoughts and feelings spontaneously.
Role play has a number of advantages, because it:
Role play can help younger students develop confidence to speak in different social situations, for example, pretending to shop in a store, provide tourists with directions to a local monument or purchase a ticket. You can set up simple scenes with a few props and signs, such as ‘Café’, ‘Doctor’s Surgery’ or ‘Garage’. Ask your students, ‘Who works here?’, ‘What do they say?’ and ‘What do we ask them?’, and encourage them to interact in role these areas, observing their language use.
Role play can develop older students’ life skills. For example, in class, you may be exploring how to resolve conflict. Rather than use an actual incident from your school or your community, you can describe a similar but detached scenario that exposes the same issues. Assign students to roles or ask them to choose one for themselves. You may give them planning time or just ask them to role play immediately. The role play can be performed to the class, or students could work in small groups so that no group is being watched. Note that the purpose of this activity is the experience of role playing and what it exposes; you are not looking for polished performances or Bollywood actor awards.
It is also possible to use role play in science and maths. Students can model the behaviours of atoms, taking on characteristics of particles in their interactions with each other or changing their behaviours to show the impact of heat or light. In maths, students can role play angles and shapes to discover their qualities and combinations.
Using drama in the classroom is a good strategy to motivate most students. Drama develops skills and confidence, and can also be used to assess what your students understand about a topic. A drama about students’ understanding of how the brain works could use pretend telephones to show how messages go from the brain to the ears, eyes, nose, hands and mouth, and back again. Or a short, fun drama on the terrible consequences of forgetting how to subtract numbers could fix the correct methods in young students’ minds.
Drama often builds towards a performance to the rest of the class, the school or to the parents and the local community. This goal will give students something to work towards and motivate them. The whole class should be involved in the creative process of producing a drama. It is important that differences in confidence levels are considered. Not everyone has to be an actor; students can contribute in other ways (organising, costumes, props, stage hands) that may relate more closely to their talents and personality.
It is important to consider why you are using drama to help your students learn. Is it to develop language (e.g. asking and answering questions), subject knowledge (e.g. environmental impact of mining), or to build specific skills (e.g. team work)? Be careful not to let the learning purpose of drama be lost in the goal of the performance.
Except for third party materials and otherwise stated below, this content is made available under a Creative Commons AttributionShareAlike licence (http://creativecommons.org/ licenses/ bysa/ 3.0/). The material acknowledged below is Proprietary and used under licence for this project, and not subject to the Creative Commons Licence. This means that this material may only be used unadapted within the TESSIndia project and not in any subsequent OER versions. This includes the use of the TESSIndia, OU and UKAID logos.
Grateful acknowledgement is made to the following sources for permission to reproduce the material in this unit:
Activity 4: © The Adventurists: http://www.flickr.com/photos/adventurists/7461888474/sizes/k/in/ photostream/, http://creativecommons.org/ licenses/ byncnd/ 2.0/; rickshaw graph: adapted from http://www.nationalstemcentre.org.uk/ elibrary/ resource/ 4252/ interpretingdistancetimegraphsa6.
Resource 1 : © Nuffield Foundation.
Resource 2: adapted from: http://www.nationalstemcentre.org.uk/ elibrary/ resource/ 4252/ interpretingdistancetimegraphsa6.
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Video (including video stills): thanks are extended to the teacher educators, headteachers, teachers and students across India who worked with The Open University in the productions.