3 Identifying variables and constants

In the previous activity the students used their experiences from real life and linked these to the mathematical concept of quantities. The second activity now moves the students on to make the distinction between variables and constants based on the quantities they identified in Activity 1.

To help students in developing their understanding of the difference between these concepts it is important that they have the opportunity to talk about it. It is very important to expect the students to use the mathematical words themselves and to create an environment where they have to do so. This will enable your students to recognise, use and communicate with one another about algebraic expressions, variables and constants. It is an important step in learning to ‘speak like a mathematician’ and to understand mathematics rather than just remembering it.

Asking the students ‘What will be the same?’ or ‘What will change?’ as in the next activity can help to trigger discussion and to identify variables and constants.

Activity 2: What will stay the same, what will change?

This activity is best done in small groups or pairs. It is important that all students have a chance to say what they are thinking and to practise their mathematical vocabulary.

  • Ask the students to look at the list of quantities from Activity 1 that they wrote in their books and listed on the blackboard. Then, in their groups or pairs, ask them to identify, discuss and make a note about which of these are:
    • quantities whose values will change, called ‘variables’
    • quantities whose values do not change, that will stay the same, called ‘constants’.
  • As a whole class, discuss with the students their findings and reasons for categorising the quantities as variables or constants.
  • Now ask your students to work in small groups, picking three of the variables and discussing what would make the variables change. Ask them to record this in their own way, but they must use the words and phrases ‘variable’, ‘constant’, ‘change’, ‘stay the same’.
  • Pick two or three of the variables that were most looked at in the previous step and invite the students to come and write down how they had recorded what would make the variables change. Discuss these records with the class, and discuss how the students think their records could be made clearer.

Case Study 2: Mrs Mehta reflects on using Activity 2

For the first part of the activity I asked the students to work in pairs or threes so they could all see each other’s writing.

I told them the question and I asked the students to draw a box around the quantities that would change. While they were doing this I walked around the classroom, but without interfering. I overheard a lot of discussions about whether a quantity would always change, or always stay the same, no matter what. At first their explanations in trying to convince each other were not always fluent, but I do think they got better at it as they tried again and again. By the time we had the whole-class discussion about the categorisation most of the students were able to express themselves pretty clearly. Those who sounded a bit muddled I asked to have another go at explaining and that helped in most cases.

We still had the list on the blackboard, and at first I also put boxes around the variables (at the students’ request) but that just looked messy. So I rewrote the list in two columns – one labelled ‘Variables – quantities whose values will change’ and the other ‘Constants – quantities whose values will stay the same’. I thought writing the mathematical terms first would help them in learning these.

They worked in groups of four to six students on the third part of the activity. I gave them a big sheet to write their records and told them we would put these on the wall at the end of the lesson (which we did). I think doing this made them think with greater precision about what they were writing. While they were working on step 3 I walked around the classroom and decided on two variables we would look at in greater detail for the last part of the activity – one was the number of wheels, and the other a rather complicated one of the number of rupees (or the price) a passenger would need to pay as their share when travelling in a group to different places.

By asking the students to come and copy their record of this on the blackboard we ended up with spider diagrams, whole sentences, algebraic expressions, and a mixture of these, all on the blackboard. What I liked about it was how it showed all these connections and different representations, and that we were able to discuss the similarities and differences in these representations.

Pause for thought

Mrs Mehta’s lesson involved quite a lot of writing and recording on the backboard, by the students themselves as well as the teacher. What do you think are the advantages and potential disadvantages of this approach?

Now think about how the activity went with your students, and reflect on the following questions:

  • What responses from students were unexpected? Why?
  • What questions did you use to probe your students’ understanding?
  • Did you modify the task in any way like Mrs Mehta did? If so, what was your reasoning for this?

2 Learning about variables and constants through talking

4 Moving on to write formal algebraic statements and expressions