1.5 Studying the Möbius band
Task 10 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 1.
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.
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.
As you worked on this task, were you simply following the instructions or was there evidence of a ‘what if?’ energy taking hold? If this is the sort of energy that you wish to encourage in your classroom, how might it 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 involv surfaces and this includes crystallography, biochemistry (for example, in work with DNA), and electronics.
As you work on the task, 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 task to someone else.
Task 11 Return of the Möbius band
Make some more Möbius bands but this time make one with 2 half twists, one with 3 half twists and one with 4 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 task, 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 task? 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, pupils 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 pupils to explain their findings verbally rather than having to write them down, while others find it easier to do annotated drawings.
Task 12 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?
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 task to pupils but also how you are asking them to record their findings. The Content/Process matrix may help you do this.
Task 13 Content and process
Look back at the Content/Process matrix. If you were using the Möbius band task with pupils 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 Task 9.
Now try the Möbius band task with someone else; it does not have to be a pupil. Watch what the person does when they are working on the task 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.
The processes involved in working on the Möbius band included problem solving, modelling, reasoning, communicating, connecting and using tools. It is a rich mathematical task because it involves a variety of processes
When you consider using a task with pupils 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. A task that has a variety of purposes was described by Ahmed (1987) as a ‘rich mathematical task’ 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 pupils 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.
You may find this to be a useful checklist when you are considering using a new task with pupils. The Mobius strip task 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 task and making it more challenging
Task 14 Stretching the Möbius band
Before reading on, think about how the Mobius band activity might be extended.
Look at your results so far from Tasks 10 and 11. Construct a table 1.1 in which you can put your results from Tasks 10 and 11. Using these results, can you predict what will happen with 5 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|
One possible extension of the Möbius band is to consider what might happen with 5 half twists, 10 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.
Task 15 The revenge of the Möbius band
Another extension of the Mobius band task might be to make the cut along a different parallel other than down the middle of the strip. For example, 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 10 half twists or 37 half twists?
A table of results has not been included this time because you may want to use the work you have done as part of your research.
While you were working on these tasks, you were asked to do some activity, talk about it and record your conjectures and findings. This triad of do–talk– record is a useful device for seeing what is going on in mathematics classrooms and is the subject of the final section of this course.