2 Contextualising equations
As discussed earlier, equations are thought of as purely symbolic expressions for most of the time. The idea is almost always to find a solution of the equation through symbolic manipulation. Each manipulation, if done in accordance with conventional rules, leads to an equivalent equation that is simpler (easier to solve) than the previous one. More meaning can be given to the algebraic notations of equations by linking them with their graphical representation, as in Activity 1.
This however is still a symbolic approach to seeing and solving equations. It does not address the concept that each equation can be seen as a representation, or model, of a real-life situation.
You can think up a story for each equation to explain it. Working on developing such stories helps to:
- enliven mathematics
- allow students to think about the mathematical processes involved in making sense of a situation
- make the students actually think about the distinction between variables and constants, and how that relationship can be changed if they decide to change your assumptions.
The next activity offers a gradual introduction to thinking of stories to contextualise equations. This will prepare your students for more complex problems later in their mathematical learning.
Activity 2: Thinking of contexts
Part 1: Changing a story
Tell your students the following:
Imagine if Mohan participated in a quiz show where he won a certain amount of money for every question that he answered correctly, with the rule that for each correct answer he gets twice the amount he got for the previous question. If, before attempting the fifth question he had already won Rs. 30,000, the equation 15x = 30,000 can be formed, where x is the amount he won to answer the first question correctly.
Can you find another context – another story – that is represented by the equation 15x = 30,000?
Part 2: Thinking of a story
Preparation
It is not important whether the students’ answers are right or wrong. Focus on (and encourage) your students’ ability to come up with a creative and imaginative context to embed the mathematics and then share how they figured out the context. A good trigger word is ‘Imagine …’
Write the equation 2x + 5 = 12 on the blackboard.
The activity
Tell your students to use their imagination to write a word problem that contextualises this equation. Ask them to share their ideas with the rest of the classroom.
Part 3: Creating equations and then thinking of a story
Preparation
This is a continuation of Parts 1 and 2, which focused on contextualising word problems. You will now ask your students to first create equations and then think of word problems that fit these equations.
Write Table 1 on the blackboard:
| 2 | 29 | 10.50 | 3/5 |
| x | 3x | 2.5x | 5x/6 |
| 5x – 8 | 34x – 12 | 5.5x + 1.7 | (2/3)x – 4/5 |
| x2 | 2x2 | x2 + 1 | x2 – 2 |
The activity
First create an equation from Table 1 by selecting a collection of expressions with the following rules:
- Two or more expressions chosen from the same row are always added and must lie on the same side of the ‘=’ sign.
- Expressions from different rows should be on different sides of the ‘=’ sign.
Then frame a word problem that describes the equation.
For example, if you chose 2 and 10.50 from the first row, and 3x and 2.5x from the second, you get:
3x + 2.5x = 2 + 10.50
5.5x = 12.50.
A sample word problem might be: ‘The area of a rectangle of length 5.5 cm and width x cm is 12.5 cm2.’
If you have access to large pieces of paper ask your students to write their problems on this paper and display it around your classroom. Ask students to walk round and read each other’s word problems. You could ask them to identify word problems that they like, copy them down and check them, giving feedback to the students who wrote them.
For more information read Resource 2, ‘Storytelling’.
Case Study 2: Mrs Mohanty reflects on using Activity 2
There was a lot of hesitation initially in doing Parts 1 and 2, as the students had never made up such kinds of questions. They needed some prompting and reassurance that any kind of story that would fit was fine; that it did not need to be plausible, or that it might help to start their story with ‘Imagine …’. For the equation 15x = 30,000, Meena said that if 15 things were bought and the total price paid was Rs. 30,000, then x would be the price of each. Sharad suggested that x could be the number of days worked and Rs. 30,000 the money earned.
Now I told them to try the one that was given in Part 2 – that is, 2x + 5 = 12. There were no takers. I decided to simply wait, without saying anything. After about 90 seconds, which seemed like a lifetime but was actually not that long, Rohit very hesitantly said if he was going to travel by autorickshaw and Rs. 5 was the minimum amount for the first kilometre and Rs. 2 for every subsequent kilometre, then x km would be the distance he had travelled. Meena at once said no, you would have travelled x +1 km. Anju came up with the example that there were two groups of children on the playground and then five more children arrived and there were now 12 children in total – how many had been in the groups? Because the answer would be 3.5 children, a discussion followed about the difference between natural numbers and non-rational numbers, which was interesting.
The discussion of Part 3 was mainly that they could form an equation easily but could not always find a realistic enough situation to describe all of them.
Like Mrs Mohanty your students may find these activities unfamiliar and need practice to become confident at thinking to word problems. Try to use this technique in more maths topics, as it will help your students to make sense of the mathematics equations.
 Pause for thought
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1 Thinking about the rules for solving equations