SUBJECT: Geometry
GRADE: High School
TECHNOLOGY: Internet
BY: Sophie Billekens

An introduction to shapes, symmetry, reflections and other frightening geometric concepts

Grade/Subject:

Geometry. This is usually given around ninth grade. Around this time students get turned off math, if they have not already been discouraged by algebra, because of proofs and abstract ideas. They begin to lose sight of where math fits in normal life. They do not look beyond the theories to why on earth they may make sense. This lesson will introduce them to the basics of geometry and try to make them see that geometry is all around them, and that it can be and is used very creatively and, very importantly, naturally.

Objectives:

• Using the World Wide Web, students will access an Escher page specified by the teacher and note and discuss the use of shapes, symmetry, transformations, reflections, and perspective in his art.

• Using the WWW, students will find their own favourite artist(s) and make notes on how their artwork uses shapes, symmetry, transformations, reflections, and perspective.

Materials:

A computer lab with Netscape or a similar browser installed in each, or at least most, computers. If not enough computers are available for each student to do the lesson individually, they can share.

Description:

The students will sit at a computer and asked to go to the following address: "http://lonestar.texas.net/~escher" This is a web page all about Escher and has quite a few pictures with little explanations of his artwork. Students will be given ten to fifteen minutes to explore this site. During this time, they ought to choose a picture they like, print it out, and write down what strikes them about the picture, why they chose that particular one out of all of them. The teacher should remind the students that they will be handing their notes and print-outs in.

Once the fifteen minutes have passed, the students and the teacher will discuss what they found interesting about the pictures. If the students do not mention this themselves, the teacher should mention what shapes Escher uses as a basis for his art, and how he uses symmetry, reflections, and rotations. Also, Escher’s “impossible” pictures should be mentioned, because they show how perspective can make something seem normal but when looked at more carefully, they do not make any sense at all. Things are all relative, and if they do not act according to basic geometric rules, they do not make sense, or simply cannot exist.

The teacher should make sure the following pictures are discussed for the reasons presented if they are not already chosen by the students:

“Horsemen” – symmetry, reflection, rotation

“Liberation” – basic shapes and their manipulation

“Reptiles” – teacher’s favourite!

“House of Stairs” – reflection, relativity, perspective

“Waterfall” – “impossible” picture

After this discussion, the students will be able to search the Web for the art of their own favourite artists. Students will look through these works and note down how shapes, symmetry, reflection, rotation, and perspective are used. They will hand in a print-out of a picture of their favourite artist and their notes jotted down. The teacher should be sure to mention that “artist” can be very broadly defined and, to her, definitely includes someone like Bill Watterson (the Calvin and Hobbes cartoonist).

Some suggested URLs are:

Da Vinci: http://www.silk.net/cia/museo.htm

Dali: http://www.webcoast.com/Dali/collection.htm

Picasso: http://picassoweb.com/posters.html

Magritte: http://www.mcs.csuhayward.edu/~malek/Magrit.html

Watterson: http://www.modeemi.cs.tut.fi/~prime/www/calvin1.html

Other artists students can look at might be Gary Larson, Vincent Van Gogh, Keith Haring, etc.

Evaluation:

Students should hand in a print-out of an Escher picture with notes on why they found the picture striking. They should note at least one of the characteristics (i.e. shapes, symmetry, reflection, rotation, perspective), if not in name then at least in general (“he uses the same shape in two different ways”). Even the not-so-obviously mathematical Escher pictures contain some of those elements in abundance. Students should contribute to the discussion. Students should hand in a print-out of a work of the artist of their choice, with notes on how shapes, symmetry, and so on are used there.