Nehru Place in Delhi, Asia’s largest market for computers and peripherals, can always become crowded. During business hours there is an extremely dynamic atmosphere. Everything from a hawker to the car park or the number of staff required in a shop is affected by how fast the environment changes from morning to evening (Figure 1). This change in an environment is called dynamics.
Professional mathematicians develop models to predict and describe these dynamics. In doing so they make it possible for urban planners, local policy makers and law enforcers to foresee what might be needed at different times in terms of labour, provisions, support structures, and so on.
This mathematical modelling relies on deciding what the variables (the numerical quantities that will vary) and the constants are (the quantities that will stay the same) in this setting. Activity 1 introduces a way to teach this with your students using an example from city life. (If your students are unfamiliar with Nehru Place or a similar environment, you could amend this example for a context they know.) The next step is to decide which variables are connected and in what way, and Activity 2 gives you an idea for how to do this with your students.
In Activities 1 and 2, you and your students will think about how to make a simplified version of such a model; note that there is no single right or wrong answer. These tasks work particularly well for students working in pairs or small groups, because this allows more ideas to be generated and students can offer mutual support when stuck.
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 learner-focused teaching environment.
Tell your students the following:
When your students have generated some ideas write the list below on the board:
Then tell your students:
On this list are some more examples of ‘players’ or ‘elements’ in this context. Between this list and your own examples, decide which are variables (with quantities that vary) and which are constants (with quantities that stay the same). Will any of these be both? If so, what would this depend on?
Activity 1 asked the students to identify variables and constants in Nehru Place. To develop a mathematical model, the students now need to think how these constants and variables influence and relate to each other.
The activity asks the students to make a mind map. A mind map is typically a series of words or phrases to represent the concept (as a node), and a line (or link) joining it to another concept, expressing a relationship of the two. A concept map is similar to a mind map, except mind maps have a centre whereas concept maps can be linear. The mind map is a good tool and provides an effective strategy to help the students explore and review their own understanding; it can also be used as an assessment tool to find out what the students know and what their misconceptions are. There are no right or wrong answers in Activity 2.
With your students, imagine again that you are professional mathematicians working on developing a mathematical model to describe the dynamics of Nehru Place in Delhi. You have already identified the variables (with quantities that vary) and constants (with quantities that stay the same) that play a role in Nehru Place.
The next step is to identify how the variables relate to each other and to the constants. To keep it manageable, each group of students should decide which four variables they will focus on. Now tell your students the following:
When the students have generated some mathematical expressions, move them into thinking out possible outcomes for their modelling. Tell them to do the following:
This is the account of a teacher who tried Activities 1 and 2 with her secondary students.
I wanted to do these activities with my students because I thought it was a lovely example of seeing and recognising mathematics in real life. We first thought of a few examples with the whole class. Straight away I asked them to sort these into variables and constants. This early discussion made them aware that this was not always easy to determine. For example, the number of car parks might be considered constants; however, if you looked at the situation over a longer time period – for example, two years – then it could become a variable because, in theory, more car parks could be built in that time if there was the space and the money.
They worked in pairs on finding more examples and thinking of reasons why and when the example was a variable or a constant. Their examples and classifications were all recorded on the blackboard. These were then used to work on Activity 2: thinking about how they relate to each other and how to record this mathematically by writing it as expressions and deciding on coefficients. A student said that she had never considered coefficients to indicate a proportion and that she now suddenly understood why there are these rules when working with expressions.
At first the students felt uncomfortable with the idea that there could be no right or wrong answers. However, after sharing some ideas about the possible expressions involving the same variables and constants, they could see why this was so and became more creative with their answers.
Because I really wanted these activities to make the students see and recognise mathematics in real life, as homework I asked them to do the same for a different situation they would encounter that night. For example, to identify variables and constants and how they related while waiting at the bus stop, having dinner with their families or doing their homework.
When you do such an exercise with your class, reflect afterwards 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 Mrs Aparajeeta did, some quite small things that made a difference.
![]() Pause for thought Good questions to trigger reflection are:
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