2 Scaling up and scaling down
Every image can be enlarged or reduced by a scaling factor. This is something we do constantly in real life. The original image and the scaled image are ‘similar’ to each other. The concept of ‘similarity of triangles’ covered in Class X often baffles students if they have never involved themselves in scaling images or have never studied the relationship between a scaled and an original image before.
In the next activity, you help students to discover their own technique for scaling a triangle. They then determine the scaling factor of the enlargement or reduction by defining the size of the enlarged triangle. This works best if the students work outside the mathematics classroom, and work on a larger scale than they are used to when working on paper. The issues of working with ‘real-life’ measurements, and the mathematical learning they gain, are described in the case study after this task and will also be explored further in Activity 3.
Activity 2: Scaling up and scaling down
This activity works well when the students collaborate in groups of threes or fours. Plan who will be in each group and why – perhaps you could mix more confident students with less confident ones. It is important that this activity takes place outside, for example on the playground, where students can write on the ground with chalk or trace drawings in soil or dust. Newspaper might be a better choice than a sheet of paper, as it is bigger. When asking students to work in groups, it is important that the resources, such as writing paper, are big enough, so they all can see at the same time. That means bigger paper sheets, and bigger writing.
Tell your students the following:
- You are given a sheet of paper. Fold it so it forms a triangle. The shape of the triangle is not important.
- Measure the triangle (angles and sides) and calculate its perimeter.
- Draw a triangle of the same shape on the ground with a perimeter that is ten times the size of your paper triangle. Is there more than one solution?
- Measure the larger triangle.
- Using the same technique as you did earlier, draw a triangle inside your paper triangle whose sides are exactly half of the corresponding sides of the original triangle.
- Compare the original triangle to any one (reduced or enlarged) triangle.
- Now think about what is the same and what is different about these three triangles.
- In your groups, without writing it down, could you think of one or more methods to describe how to get from one triangle to the others? Check whether the methods would also work for starting with one of the other triangles.
- Now write down your method.
Look at Resource 2, ‘Using groupwork’, to find out more.
Case Study 2: Mrs Mohanty reflects on using Activity 2
The groups started to discuss [the activity] but there were several students who were not contributing at all. I realised that they were not finding the task easy, but I did not really understand what it was they were stuck on. The first part of the question asked them to make a triangle out of paper – surely, I thought, that is not that hard. I began to think that perhaps they had forgotten what they were supposed to know already about triangles and maybe I should ask them to quickly look in their textbooks to refresh their memory.
However, we were in the playground, and the books were in the classroom, so that would take too much time. So I thought, ‘OK, I actually do not know what they are stuck on – maybe I should simply ask!’ At the same time I thought, ‘I actually don’t like it when they are stuck.’ They always seem to wait for me to help them become unstuck by telling them what to do. They expect me to break down the questions further into mini-steps. I don’t think that helps them become good independent learners or problem solvers, which is what we need them to be in life outside school. So I said exactly what I was thinking:
You all seem to be stuck. Being stuck is part of life. What is important is to become unstuck and know how to become unstuck. So could you please first discuss in your groups what you are stuck on, and how you could become unstuck. We will then discuss some of your ideas with the whole class.
It took us less than five minutes, and they had some good ideas about becoming unstuck. The discussion actually gave us a sense of togetherness within the whole group – a sense of joint responsibility to help each other with learning, and getting unstuck, and the realisation that we all can do that. There were lots of smiles for the rest of the lesson. As a result, later on, we were able to discuss similarity and also congruency, scale factor and the properties of all triangles. So rather than learning just one aspect of mathematics, they started to connect different aspects.
The question asking them to construct a similar triangle with ten times the perimeter caused some head-scratching – but not frustration. I overheard students saying, ‘We are stuck’ and ‘Now we have to become unstuck.’ One group asked for string to get unstuck. I had not thought of that, so had not brought it. Luckily I had a bobbin of strong sewing thread in my classroom that they happily used.
The question asking what was the same and what was different was interesting. I could see the students physically walking around their triangles, looking at them from different angles. They voiced all kind of conjectures, thought again and then rephrased their conjectures. It led automatically to describing a method. I think it was good that they did not write their ideas down immediately, because they seemed more willing to come up with different descriptions, and could go at a faster speed – writing always seems to slow things down. They also had to become more precise in their use of mathematical language, because otherwise the others could be standing looking at the triangles from a different place, and getting confused when a student said ‘that angle’.
They wrote first on a big sheet of paper, with diagrams, and then copied their method into their exercise books. Then of course I did not know what to do with those big sheets of paper! Eventually we stuck them on the walls of the classroom. It did not look particularly neat; however, it looked exciting, and it was an example of their mathematising. We called it the ‘thinking work wall’. I noticed that over the days and weeks, the students, and other teachers, would read and discuss it. I really liked that there was still talk about a lesson and the learning that happened long after the lesson was over! The same thing happened with the chalk drawings in the playground: people were intrigued and talked about it. The wall display stayed up until we replaced it some weeks later with different ‘thinking work’ from the students.
The activity made me reflect on the effect that making mathematical constructions outside can have. Outside they work on a larger scale than they can in their exercise books, but why would I bother doing this with my students? I still have not figured it out completely why it seems so useful, and I doubt I ever will, but these are my thoughts:
- It was big enough so they all could see what was happening at the same time and take part in the discussion, truly collaborating.
- They had to work together in constructing the triangles. They could not do it on their own because it was too big and there were too many things to do. Because of that, they had to communicate and talk about mathematics. There was also a sense of joint responsibility for the learning.
- It did not have to be precise and accurate from the word go, and that allowed the students to explore more and try out more ideas.
- Their world of mathematics was literally enlarged. The students went from having to sit down in the classroom with a textbook, into their ‘outside’, ‘real’ world of the playground, where they hang out with their school friends.
What also struck me was how full of joy and excitement the lesson had been, together with deep thinking and mathematical contemplations: the smiles and lit-up eyes when they came up with a conjecture, and then the realisation that their conjecture actually held true (at least for a while!). They gleamed with pride and a sense of achievement, and so did I!
Pause for thought