2 Using substitution to think about possibilities
Substitution is a powerful and necessary thinking tool. In real life, substitution is used constantly: for example, deciding which different herbs and spices to use in today’s meal to make it different from yesterday’s; which mode of transport to use (will you walk, or take a rickshaw or bus?), or which sari to wear. Substitution involves thinking about possibilities and alternatives whilst also considering limitations. For example, substituting a silk sari for a calculator will not be very valid when the aim is to change your dress, but could be valid when considering what you are going to spend your money on. Substitution thus allows for variation – the spice of life.
The thinking process of substitution in mathematics is similar to that in real life. It involves considering examples, alternatives and possibilities, while at the same time being aware of limitations and restrictions.
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Learning opportunities when working on substitution
For their studies in mathematics, students have to learn the skill of substitution – many exercises and problems in exercise books and examination require students to demonstrate this skill.
Giving students problems to solve that involve substitution also offers other opportunities for learning mathematics that are perhaps not so obvious but which are equally important and worthwhile. Some of these, which will be explored further, are:
- Substitution as a tool to make mathematical expressions come alive by making a story of what an algebraic expression can mean. Activity 3 is an example of this.
- Substitution as a tool to generalise a specific numerical example. Activity 4 is an example of this.
- Substitution to explore mathematical relationships, properties and limits. Activity 4 has some examples of this.
Using substitution to develop imagination
Algebra and its expressions are often considered to be the language of mathematics because they describe relationships. When doing page after page of practice from the textbook on manipulating algebraic expressions, students might be forgiven for not recognising this as eloquent language!
The next activity aims to introduce some playfulness in reading algebraic expressions and asks students to come up with their own stories of what the expression might mean and what situation it models. This means that the students are asked to give meaning to the symbolised mathematics by substituting the symbols with a narrative or story. For example:
- The expression + 3n could be about a group of people (x) who in equal numbers boarded a number of buses (n). There were some people who arrived late and each bus had to accommodate another three people.
- The expression 1.3(2100m) could have come from the expression 1.3 [3(700)m] and be about 700 space aliens who landed in Udaipur. This was their first time in India. They saw that people seemed to like a drink called chai which cost Rs. 1 per cup. They tried it, and loved it so much that they had three cups each. When they wanted to pay for the tea they were so pleased with the quality of the chai and the service they had received that they paid a tip of 30 per cent.
Activity 3: What could it mean?
Tell your students the following.
Below are some algebraic expressions adapted from textbooks:
- 30u + 44v
- 3x +1
Use your imagination and make up a story for what each expression could be about. What situation could the expression be modelling? What else could it be about?
You may also want to have a look at the key resource ‘Storytelling, songs, role play and drama [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’.
Case Study 2: Mrs Kapur reflects on using Activity 3
Thinking of an actual story to fit an expression was not something I myself found easy. But I found that the examples given about the people on the bus and the aliens drinking copious amounts of chai tea really helped me and made me think more freely and creatively of other examples. I decided that because this had worked well for me, I would give it a go and use the same approach with my students. So I started the activity by writing the expressions for the bus and alien stories on the blackboard, giving them a few moments (not too long) to think of a story and then sharing it with the class.
I then wrote the expressions of Activity 3 on the blackboard. Rahul immediately thought that the expression 3x + 1 could tell the story of how much it cost to make three cups of tea: x rupees for the water and tea, and a fixed cost of 1 rupee for something. Some students agreed and others did not, but an interesting discussion followed. It made me realise that working from a somewhat imperfect example is actually more effective in generating discussion than having a perfectly correct answers.
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1 Variables and constants in a real-life context