4 Task design to help students think about mathematical properties
In Activity 2, students constructed their own mathematical ‘truths’ by building on their natural powers for deductive reasoning and using their knowledge of mathematical properties. Knowing about mathematical properties is very important, because any argumentation in mathematical proof should be based on these. Similarly, defining mathematical conditions about when a statement applies and when it does not is equally important.
Sorting and classifying tasks are effective in supporting students to become aware of defining properties and conditions for mathematical sets; that is, which properties belong to which set. Sorting and classifying makes you think about differences and similarities. The next activity uses this approach.
Activity 3: Sorting and classifying to find properties and conditions
Tell your students that a ‘proportion’ is a pair of ratios that are equal. Being proportional is a property that occurs frequently in mathematics across different topics and concepts.
The statements that follow describe some scenarios that illustrate mathematical concepts. Ask your students to read these and classify whether they would be proportional or not, and under what conditions. They should provide their reasons and explain to a friend what the example would be proportional (or not proportional) to. Be critical!
- a.Enlarging a photograph.
- b.Cutting slices of bread to make a sandwich.
- c.The equation of a linear equation in one variable (ax + b = 0).
- d.The work done by some number of people in a given time frame.
- e.The amount of money spent by a number of people to watch a particular movie at a particular cinema.
- f.The fare for tickets for train journeys.
- g.Dividing a line segment into m equal parts.
- h.The height of Qutab Minar (Figure 1).
Figure 1 Qutab Minar, Delhi.
- i.The area of a circle.
- j.The area of a square.
- k.The product of the height of a cylinder and its radius.
- l.The volume of a cone.
- m.A rickshaw fare.
Case Study 3: Mrs Agarwal reflects on using Activity 3
To help the students engage with this activity, we read the instructions and examples (a) and (b) together aloud in the classroom. We then had a brainstorm about possible answers for (a), and discussed the conditions that would need to be fulfilled for the enlarged photograph to stay proportional to the old one. This was started by Rahul, who said that to enlarge the photograph to make it twice as big, you would have to double its width. I could see that Usha was thinking, as she was frowning, and then she said that would not work because it would make the picture look weird.
I then asked the students to first work on the examples individually so they could develop their own thoughts and ideas first, and then to discuss their thinking and reasoning with their partner.
After some time, I picked some statements to discuss with the whole class and for the students to share their ideas. I noticed that students were using some phrases repetitively, such as:
- ‘But what if …?’
- ‘If you change this, then that changes/still stays the same.’
- ‘This is the same, this is different.’
I pointed this out to the students and they agreed that these were good phrases to find out about mathematical properties and conditions. Two students quickly wrote the phrases and a title ‘Good phrases to find out about mathematical properties and conditions’ on a large piece of paper, which was then stuck on the wall.
 Pause for thought
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3 Coming to know mathematical properties and facts