1 Variables and constants in a real-life context

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.

Figure 1 Dynamics in real life: Nehru Place, Delhi, when it is quiet (left) and busy (right).

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.

Activity 1: Identifying constants and variables

Tell your students the following:

  • Imagine you are a professional mathematician and you are working on developing a mathematical model to describe the dynamics of Nehru Place in Delhi. You first have to identify all the variables (quantities that vary) and constants (quantities that stay the same) playing a role in Nehru Place.
  • Make a list of all the ‘players’ or ‘elements’ in this setting. Some examples could bethe car park, the hawkers or the number of shops on the first floor.

When your students have generated some ideas write the list below on the board:

  • The number of:
    • police men and women who work at the police department in charge of security at the complex
    • car parks
    • people employed by the municipal corporation that is in charge of civic maintenance of the complex
    • parking lot attendants
    • hawkers
    • escalators
    • shop owners whose shop is on the first floor
    • restaurant owners on the ground floor
    • electricity supply companies
    • visitors wanting to purchase a laptop.

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.

Activity 2: Developing algebraic expressions

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:

  • Make a mind map of these variables and write on the lines connecting the ideas how you think they might relate. Add some constants to the mind map if you think they play a role in the relationship. Remember there are no right or wrong answers for this! For example, you could think that the number of police officers should vary depending on how many visitors (buyers) there are at any given time, or on the number of shops or cars.
  • Now decide which quantifiers you would use in the relationships your group described above. Write these as a mathematical expression. For example, you could state that you would need one police officer for a combination of every ten shops, 100 visitors or 50 cars; in which case you could write a model of the number of police officers like this: s/10 + v/100 + c/50. Remember there is no right or wrong answer!

When the students have generated some mathematical expressions, move them into thinking out possible outcomes for their modelling. Tell them to do the following:

  • Predict the range of values for each variable. In cases where you are having difficulty predicting a range, identify the reasons for the difficulty. For example, the number of escalators cannot be less than one, because you cannot have half of an escalator. You also cannot have an unlimited number of escalators, because they take up space. Deciding on the maximum number of escalators is harder to do because it will depend on several factors.
  • Decide which of the variables you think can be controlled easily? Controlling a variable could mean either that its range can be restricted or that its value can be fixed without affecting the situation very much.
  • At the end of the activity, ask the whole class to discuss this point: in reality, the quantifiers used in modelling will be based on data. If you had to organise this, how could you collect the information?

Case Study 1: Mrs Aparajeeta reflects on using Activities 1

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.

Reflecting on your teaching practice

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:

  • How did it go with your class? Did all the students participate?
  • What responses from students were unexpected? Why?
  • What questions did you use to probe your students’ understanding?
  • Did you feel you had to intervene at any point?
  • What points did you feel you had to reinforce?
  • Did you modify the task in any way? If so, what was your reasoning for this?

What you can learn in this unit

2 Using substitution to think about possibilities