Can these constructions actually exist?
The aim of this task is to argue, conclusively, about whether or not the following shapes and diagrams can actually exist! To be successful, you will have to use what you know and can deduce about the geometry of triangles and intersecting lines! It is a simple task to understand, but might not be so simple to achieve. It is not enough to have a 'feeling' about whether or not these could exist, you have to be able to justify it with a reasoned argument! This should be a great opportunity for debate in the classroom.
A quick sample!
The artist, MC Escher, had a real interest in images of 'impossible diagrams'. The idea that something can be drawn and give the appearance in that drawing of being real, when in fact, the drawings could not exist, as they were depicted, in reality. This is an interesting thought worth reflecting on - we can draw pictures of things that could not physically exist. The Penrose stairs on the left are a great example. As they are drawn, it seems that you can be perpetually moving either up or down the stairs. These made an appearance in Christopher Nolan's mind bending 2010 film 'Inception'. In the clip below the character says 'In a dream you can cheat architecture in to impossible shapes' - Open the hidden box to watch the clip.
Any time spent searching for Escher's optical illusions will give plenty more examples of these kind of 'Impossible diagrams'. The key point is that - in the study of mathematics, we have to be discerning, we can't accept things that 'look like' they might be true, but we have to be able to justify with logical reasoning why they are or are not! This exercise is all about just that. You have to look at all the information you have including diagrams, numbers and symbols and use this information to draw your conclusions.
You can start this activity straight away using the Impossible Diagrams activity sheet, embedded below.
Its a good idea to think about the sorts of things that students might say when asking questions and making reasoned arguments with each other. The following are some examples!
- Teachers might introduce the activity by showing some of Escher's impossible diagrams. This can lead to some excellent discussion and arguing that sets the tone nicely for the rest of the activity.
- It may be useful to discuss or demonstrate what is meant by a reasoned argument. Perhaps the teacher might start by leading a whole class through one example. Alternatively, once the activity is started, the teacher might invite a student or group of students to share their arguments so that the process can be refined by using an example.
- The diagrams can be given out as a whole activity sheet or one at a time on larger sheets to pairs or groups. This depends how the teacher wants the lesson to go. This might also work well as a 'Silent debate' where the examples are laid out on a large table with large sheets of flip chart paper or similar. Station students at different starting points and ask them to annotate the diagrams with their arguments. Gradually move students around the table so they can contribute to other diagrams. You can differentiate here by offering different starting points and giving students different colours.
- There are lots of ways you could run it, but it is important to take the opportunity for students to articulate and share their arguments for the rest of the group.
- The aim would be to end with a unanimous agreement about which are possible and which are not on the strength of the students arguments.
- These can then be checked against these answers