2 Developing an understanding of perimeter
Mathematical vocabulary is not always straightforward and may act as a barrier to learning. It is often helpful for students to appreciate this, and for the teacher to draw special attention to mathematical words and where they come from. Greek students do not tend to find the word perimeter difficult to understand because the word comes from the Greek words peri (which means around) and meter (which means measure).
In the first activity you will ask the students to explore perimeters by describing, tracing and working out the perimeter of everyday objects. Then you will ask them to use this knowledge to explore possible variations in drawing different rectangles with the same perimeter, and to generalise their observations.
Before attempting to use the activities in this unit with your students, it would be a good idea to complete all, or at least part, of the activities yourself. It would be even better if you could try them out with a colleague, as that will help you when you reflect on the experience. Trying them for yourself will mean you get insights into learners’ experiences which can, in turn, influence your teaching and your experiences as a teacher.
When you are ready, use the activities with your students and, once again, reflect on how well the activity went and the learning that happened. This will help you to develop a more student-focused teaching environment.
Activity 1: Finding perimeters of objects that surround us
In preparation for these tasks, ask your students to point out, and if possible trace around, the perimeters of several objects they can see in the classroom. Discuss with them the mathematical definition of perimeter, which is the path around a two-dimensional shape.
The students work in pairs. Ask them to find the perimeter of at least three objects they can find in their bags and around the classroom. Give them a time limit (for example four minutes). Stand back and observe – there is no need to interrupt, or give more hints. To help your preparation for the pair work you can use Resource 2, ‘Managing pairs to include all’.
At the end of the time limit ask the students for feedback. Because they will have worked out the perimeter of different objects, they will not all have the same results.
Now ask the students to feedback some information about the shape of the object they found the perimeter of, and then their method of finding it out. Write these answers on the blackboard (leave these answers on the blackboard – you can use these in Activity 2), or ask the students to come and write them on the blackboard.
For this part of the activity it is helpful to have squared paper for the students to work on.
The students continue to work in pairs. Ask the students to draw as many rectangles or squares that they can think of with a perimeter of 16 and to prepare to answer the question, ‘How do you know you have got all solutions?’
Take feedback about possible solutions and how they know they have got all possible solutions. Try not to tell the students the reason (two numbers that can be added together to make six), but try and let the students formulate this observation.
Video: Monitoring and giving feedback
Case Study 1: Mrs Aparajeeta reflects on using Activity 1
This is the account of a teacher who tried Activity 1 with her elementary students.
When I asked the students to point out perimeters and areas in the classroom I was surprised that they did not say ‘this is the perimeter of the door’ and point at the edges of the door. What happened was that a few students explained how to calculate the perimeter and others looked a bit bewildered.
I really needed to prompt them, and give an example myself before they could say ‘this is the perimeter of the door’ or ‘the perimeter of the blackboard would be this’ and use their hands and fingers to indicate and point this out. By spending some time on this, the other parts of the activity went very smoothly and I got the impression that most of the students now understood what they were talking about and what they were finding out, and understood better their methods for finding out the perimeter.
When Part 1 of the activity was given out they were all very enthusiastic. They took some items from their bags to find the perimeters. One brave student, Dheeraj, was trying to find the perimeter of his pencil. He got hold of a thread and tried to wrap it round the pencil to get the answer, so I asked him to note down the difficulty he had in doing this and that we would discuss this with the rest of the class.
Then we had a lively discussion about the items they had found the perimeters of, and how they had gone about finding the perimeter. At that point I asked Dheeraj to share his predicament with the rest of the class. So then while discussing it, it came out that perimeter is something they can find for two-dimensional things and so we had more discussion about dimensions and solids and if we had been working with solids then what could we find the perimeter of (different faces, different cross-sections, etc.). I was amazed by this discussion, not only because of the mathematics that we discussed but also by the students’ ability to express themselves and come up with mathematical ideas and theories themselves – even those who are usually shy and quiet.
Drawing the rectangles with a fixed perimeter was really fun for the class. They did this very quickly. Some did make a mistake because they added just two sides to get 16 cm and so there was a great discussion amongst them of how their perimeter was not 16 cm but 32 cm. As for whether they had got all the options in Part 3, this was explained by Shanu very well. To also get the other students involved in that discussion I asked them whether they agreed with Shanu, understood the reasoning, and could explain it in another way.
Reflecting on your teaching practice
When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and able to progress, and those you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting as Mrs Aparajeeta did, some quite small things that made a difference.
Pause for thought
In Mrs Aparajeeta’s lesson, Dheeraj’s attempt to find the perimeter of the pencil led to some discussion that went beyond Mrs Aparajeeta’s original plans for the lesson. What do you think are the advantages or disadvantages of allowing students’ discussion to move in a different direction? What might be the implications of this for planning future lessons? Now think about how your own class got on with the activity and reflect on the following questions: