3.1 The Möbius band
Simply reading words on a page does not mean that you have necessarily engaged with the mathematical ideas or done any mathematical thinking. In order to help you work on the ideas in a ‘hands-on’ way, you are asked to model the conveyor belt problem using a physical model.
Activity 11 The Möbius band
Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then tape it together. This is a Möbius band as shown in Figure 2.
Check that the Möbius band has just one face by using a pencil to mark down the centre of the strip – it meets up with itself!
Look at the band and imagine cutting down the middle of it, along the pencil line. Can you see what is going to happen? Make the cut.
Repeat this process for a second cut down the (new) centre of the strip.
Comment
Since you have already read the article, it may have been obvious to you what would happen. But many adults and children actually need to see it happen in order to believe it. After the first cut, you should have a band that was twice as long as the original and half the width. The second cut produces two connected bands of the same length but half the width – a less obvious result.
A useful general strategy for stimulating pupils’ curiosity and channelling the energy that results is to invite them to pose ‘what if?’ questions. As an illustration, when you worked on the Möbius band in Activity 11, some suggestions were provided in these activities for things to try. A more open-ended (and in many ways more satisfactory) version of this activity would be to ask you to pose a series of ‘what if…’ questions for yourself. For example, what if I used two half twists/three half twists, and so on. Would you have welcomed more openness in this activity? For example, were you simply following the instructions or did you start to feel that there was evidence of a ‘what if?’ energy taking hold within you? If this is the sort of energy that you wish to encourage in your classroom, it is worth considering how it might be fostered?
You have now tried out the basic idea of a Möbius strip but it can be expanded further and this involves more cutting and sticking (doing), and more talking and recording … but not always in that order.
The Möbius band was discovered, in the nineteenth century, by the German mathematician and astronomer Augustus Ferdinand Möbius (1790–1868). The Möbius band is a standard problem type in an area of mathematics called topology – a branch of geometry concerned with the properties of a figure that remain unaffected when a shape is distorted in some way (perhaps when stretched or knotted). Topology has applications in contexts that involve surfaces and this includes crystallography, biochemistry (for example, in work with DNA), and electronics.
As you work on the activity, make notes about how you work on it and the discoveries you make. Note down any predictions or conjectures you make and remember to record your findings. Also think about the processes you have used and how you might present the activity to someone else.
Activity 12 Return of the Möbius band
Make some more Möbius bands but this time make one with two half twists, one with three half twists and one with four half twists. It is a good idea to label them at this stage or perhaps use strips of different colour.
Note down how many faces each band has and then cut each one down the centre line.
Before you cut, remember to predict what you think will happen.
While you work on the activity, explain to yourself or to someone else what you are doing and what you are thinking.
Record your predictions and results.
How did you feel when you were doing this activity? Both adults and children often report being very excited by the results but also pretty baffled. It is sometimes difficult to see what is happening with some mathematical problems, and even trickier to predict what is going to happen. Many people record their findings and conjectures in a fairly haphazard way but even so they can usually retrace what they did. When working on an investigation it is common for people to make jottings rather than organised notes. The advantage of this form of note taking is that it does not slow down the investigation.
However, if you wanted to explain your findings to someone else you might reorganise your notes in a way that someone else could follow. Too often, learners can get caught up with the presentation of the work rather than the exploration of the mathematics. This can result in a loss of creativity and of a sense of purpose and enjoyment. It also is easier for some learners to explain their findings verbally rather than having to write them down, while others find it easier to do annotated drawings.
Activity 13 Recording
Look back at the way you recorded your findings. Was it ordered, apparently haphazard, neat and tidy? What was the purpose of your recording?
Comment
The purpose of your recording was at least threefold. The first was that you had been asked to do it and this recording could form part of a TMA. The second was that recording may have helped you to keep track of what you were doing. The third possibility is that the notes you made, and the way you recorded them, may have helped refine your predictions and conjectures.
As a teacher it is important not only to reflect on why you are offering a particular activity to learners but also how you are asking them to record their findings. The content/process matrix may help you do this.
Activity 14 Content and process
Look back at the content/process matrix (Figure 1). If you were using the Möbius band activity with learners, which cells do you think you could fill in? You may find the list of mathematical processes useful. You may also wish to refer to the list of processes you identified in Activity 10.
Now try the Möbius band activity with someone else; it does not have to be a learner. Watch what the person does when they are working on the activity and ask them to explain their thinking as they work. Pay particular attention to the process skills that they draw on.
Look back at the matrix and see if you can fill in any more of the cells.
Comment
The processes involved in working on the Möbius band included problem solving, modelling, reasoning, communicating, connecting and using tools. It is a rich mathematical activity because it involves a variety of processes.
When you consider using an activity with learners it may help you to consider whether it has a limited number of purposes or it could be used in a wide variety of ways. An activity that has a variety of purposes was described by Ahmed (1987) as a ‘rich mathematical activity’ being one which:
- must be accessible to everyone at the start;
- needs to allow further challenges and be extendible;
- should invite children to make decisions;
- should involve children in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting;
- should not restrict learners from searching in other directions;
- should promote discussion and communication;
- should encourage originality/invention;
- should encourage ‘what if’ and ‘what if not’ questions;
- should have an element of surprise;
- should be enjoyable.
(Ahmed, 1987, p. 20)
You may find this to be a useful checklist when you are considering using a new activity with learners. The Möbius strip activity can be used in a way that fulfils many of these criteria. You are now asked to explore the second criterion in Ahmed’s list by extending the initial activity and making it more challenging.
Activity 15 Stretching the Möbius band
- Before reading on, think about how the Möbius band activity might be extended.
- Look at your results so far from Activities 11 and 12. Construct a table similar to Table 1 in which you can put your results from Activities 11 and 12. Using these results, can you predict what will happen with five half twists, and so on?
| Number of half twists | Expected result | Actual result |
|---|---|---|
| 0 | 2 strips, same length as original, half the width | |
| 1 | 1 strip, twice the length of original, half the width | |
| 2 | 2 linked strips, same length as original, half the width | |
| 3 | 1 strip, twice the length of original, half the width | |
| 4 | 2 linked strips, same length as original, half the width |
Comment
- One possible extension of the Möbius band is to consider what might happen with five half twists, ten half twists, 37 half twists? However, there are clearly problems that might emerge in continuing to do this as a practical activity.
- You need to be able to predict what will happen rather than continuing to make and cut the bands. In general, setting out results in the form of a table is a useful way of investigating patterns in your results.
The layout of the table can sometimes help you to see the pattern, or the general case. From the comments on this table it looks like the strips with an odd number of twists behave in a different way from those with an even number.
Activity 16 The revenge of the Möbius band
By using your own ‘what if…?’ question, can you think of another extension of the Möbius band activity?
Comment
Another extension of the Möbius band activity might be to ask what might happen if you make the cut along a different parallel other than down the middle of the strip. Alternatively, you could investigate what happens when you make the cut one-third of the way along each band.
Before you make each cut, take time to predict what will happen. Make your conjecture and record it. Also record any surprises or questions that you have.
Can you predict what will happen with ten half twists or 37 half twists?
While you were working on these activities, you were asked to do some activity, talk about it and record your conjectures and findings. This triad of DTR is a useful device for seeing what is going on in mathematics classrooms and is the subject of the final section of this course.