4 Moving on to write formal algebraic statements and expressions
Professional mathematicians develop models to predict and describe dynamics and changes in what is happening. In doing this they make it possible to foresee what might be needed when changes happen, which is very important in all planning. This mathematical modelling relies on deciding what the variables and constants are, which ones are connected and how they are connected. This has been considered in Activities 1 and 2. The next step is to decide how these variables and constants influence and relate to each other and to record this ‘model’ in a mathematical way by making mathematically expressed statements.
The next activity will develop your thinking about how to make simple versions of such mathematical models, and build on the learning from Activities 1 and 2. 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 they are stuck.
Activity 3: Variables and constants in algebraic expressions
[Note for the teacher: this task can be simplified by using whole numbers only.]
Part 1: Mr Murti calculates his autorickshaw fare
- Remind the students of Mr Murti and his travels around the city in an autorickshaw. To make sure he does not get overcharged, Mr Murti likes to calculate how much he has to pay himself.
- The autorickshaw driver charges Rs. 25 for a journey that is up to 2 km. After that the fare is Rs. 0.80 for every extra 0.1 km.
Ask the students to do the following:
- Calculate the fare that Mr Murti pays for travelling 3.6 km, 6.7 km, 12.3 km, 25.9 km, 31 km, 1,000 km, 1 crore km, and finally ‘x’ km.
- Write down the way that you worked out each answer. Did you need to change your method to find the cost for x km? Check your answers with your classmates.
- Now complete this statement in algebra or in words:
If Mr. Murti travels x km, he will have to pay a fare of ___________.
Note that the whole sentence above, ‘If Mr Murti …’, is called a statement. What the students have to fill in at the end, in this case something similar to 25 + 8(x – 2), where x > 2, is called ‘the algebraic expression’. If students cannot write it yet in algebraic notation they should write it in their own words in a sentence.
Part 2: Make up your own statements
Ask the students to use the list of variables and constants they made in Activity 2 to construct their own statements in words or with algebraic expressions using variables and constants for concepts like cost or time.
Your students will probably not all be at the same stage in their understanding of how to construct their own statements using variables and constants. This activity should provide you with an excellent opportunity to monitor their performance and provide them with constructive feedback. You may wish to have a look at the key resource ‘Monitoring and giving feedback [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’ to help you prepare for this aspect of the activity.
Video: Monitoring and giving feedback
Case Study 3: Mrs Aparajeeta reflects on using Activity 3
I asked the students to work on part 1 of Activity 3 in pairs or in groups of three because I thought that might help them to get more ideas and to get un-stuck if they did not know how to do it.
While they were working on this, I walked around observing how they went about working out the fare Mr Murti would have to pay. I noticed that they used different methods. I thought it would be nice to share those with the whole class so the students could see that there were several ways of solving a question. So after about five minutes I stopped the class and asked two students that I knew had used different approaches to come to the blackboard and write them down. I then asked who had done it differently and asked them to explain how they had done their calculations.
I noticed that not all students were listening, so I then asked all the students, in their pairs or groups, to find justification for the methods others had used, and then we shared this again with the whole class. This led them to discover and discuss misunderstandings. For example, Seema and her partner had found multiples of 0.8 km and the costs for each of those multiples. They had then used the one closest to what was asked in the question. She had, however, forgotten that the first two kilometres was a fixed price. Jai’s group examined her method and pointed out that they had forgotten this fixed price. So what then happened was, not only could they see the different ways of getting to a solution, but they could also see what they had missed.
Most of the students were able to complete the statement in words, and about a third of the class attempted to then complete the statement with an algebraic expression. The algebraic expression was seldom correct however. I asked a couple of students to come and write their algebraic expression, together with their statements, in words on the blackboard. We then discussed how they were linked, and whether we could improve on the mathematical notation.
Part 2 of the activity helped them in experimenting with this more, which was useful. I think for the student to be able to talk about algebraic notation and how it relates to a statement in words was really helpful and I realised I had actually never given them a chance to do that before.
Pause for thought
3 Identifying variables and constants
5 Putting algebra into cricket