Model Unit (Grade 7)
M.C.Escher: The Poet of the Impossible

In this unit, we see integration of two subjects (mathematics and visual arts) with the concrete (the art of M.C. Escher) . Although it addresses the Grade 7 level, it could easily be modified and adapted to respond to the needs of students at other grades. Throughout the unit, suggestions and additional activities have been provided to facilitate these modifications and adaptations. Still other suggestions have been included at the end of the unit.

There are several levels and ways of achieving integration. The following unit is a model only; hence, teachers can use other methods of integration. It is in fact recommended that teachers experiment with a number of methods for carrying out integration within mathematics, with other subject areas and with everyday life.

General Description of the Unit

Following the description of the stages you will find suggestions for adapting the activities, additional strategies, a magazine article, measurement instruments for evaluation, references, and worksheets.

Unit Overview

Overview

After they have studied examples of tiling encountered in everyday life, the students explore tiling from the perspective of mathematics and visual arts through studying the works of M.C. Escher.

Learning Objectives

The student will:

• increase his/her knowledge of two-dimensional shapes through the study of their attributes (sides and angles) (Mathematics);

• develop facility in using mathematical language (COM, NUM);

• expand his/her knowledge of geometric transformations (Mathematics);

• increase knowledge of tessellations (Mathematics);

• collect data, organize it in the form of a table, and draw conclusions from it in order to solve a problem (Mathematics);

• solve problems (NUM, CCT);

• understand that there can be more than one way to solve a problem and that there can also be more than one acceptable solution (NUM);

• identify a variety of sources of inspiration and start to consider them as a point of departure for a personal expression (Visual Arts);

• recognize the role and the influence of visual imagery in everyday life (Visual Arts);

• respect others through cooperative learning (PSVS);

• foster intuitive and imaginative thinking that goes hand in hand with the ability to evaluate ideas, procedures, experiences, and objects in a meaningful context (CCT);

• use computers as learning tools (TL).

More specifically, depending on the activities chosen, the students will be able to:

• design a plan and solve problems using a table (P-2);

• solve a variety of problems (P-4);

• display the results using a variety of means (P-11);

• measure angles (G/M-6);

• recognize that the sum of the interior angles of a triangle is equal to 180°, and the sum of the interior angles of a quadrilateral is 360° (G/M-14);

• identify, name, give an example, and list the properties of figures (G/M-13);

• combine 2-D geometric figures (polygons) to make other figures (2-D or 3-D) (G/M-14);

• understand and use the terms congruent, congruency, tessellate, tessellation (G/M-16);

• identify the relationships between the sides and between the angles of similar polygons (G/M-17);

• continue or create patterns using translations, reflections, and/or rotations (G/M-33);

• recognize tessellations in the environment and tessellate a surface by using one or more shapes (G/M-34);

• explain why some shapes tessellate and identify and create shapes that tessellate (G/M-35);

• determine which properties are preserved by translations, rotation, and reflections (such as congruency, area, orientation) (G/M-36);

• acquire data (D-1);

• discuss, interpret, and ascribe meaning to the organized data (D-14); and,

• read various charts and schedules, and use the information gained to solve problems (D-18).

The range of methods used can be broadened according to the adaptations made or the additional strategies pursued.

Methods of Evaluation

The following measurement instruments have been appended:

• anecdotal report sheets;
• observation charts; and,
• rating scales.

Other suggestions are offered throughout the unit.

Advance Preparation

Locate the recommended resources. You can access the resource centre at the school or the school board, the public library, parents, or colleagues.

Have accessible scissors, adhesive tape, pencils, paper, coloured pencils, paint, etc., so that the students will not have to spend time looking for supplies.

If the teacher has never before engaged in this kind of activity, it is recommended that he or she do so before starting the unit with the students. The students will then already have some models for reference. It is also a good idea for the teacher t o show students how to carry out these activities during the unit.

Resources and Materials

• Pattern blocks, attribute blocks, and/or tangram-tiles.

• Protractors.

• Works of art by M.C. Escher.

• Examples of paving and tiling found in magazines, books, or the environment: patterns used in flooring, patios, wallpaper, fabrics, quilts, stained glass windows, mosaics, ornamentation from different cultures, works of art.

Vocabulary and Structures

In the course of this unit, the student will be encouraged to understand and use:

• the vocabulary and the concepts associated with mathematical activities: covering a surface, paving an area, laying tiles, mosaics, paving, tiling, tessellation, two-dimensional shapes, polygons, triangles, quadrilaterals, trapezoids, rectangles, squares, pentagons, hexagons, octagons, congruency, reflection, rotation, sliding, turning, flipping, angles; and,

• the vocabulary and the concepts associated with visual arts: evenly filling an area, rhythmical repetition of a pattern, decorative art, ornamentation, paradox, metamorphosis, an impossible world, drawing from observation.

Start-Up

Present the students with examples of paving, tiling, or tessellations. Examine a number of these paved surfaces and point out which shapes are used in creating them.

By studying these techniques you can develop a definition of paving, tiling, or tessellation.

Give each student a pattern block and a worksheet and ask the students to cover the surface or pave the area so as to produce a tessellation. The teacher can demonstrate at the same time using the overhead projector.

Point out the fact that there are many ways to cover an area using the same shape. Examples 1 and 2 show several methods employing a square.

In this way, art and mathematics can be integrated to produce a number of different results.

Remarks

These examples can be taken from magazines or books or be found in the environment: patterns used in kitchen or bathroom floors, patios, honeycombs, wallpaper, fabrics, quilts, stained glass windows, mosaics, ornamentation from different cultures, work s of art.

Tiling, paving, or tessellation is defined as covering a surface or an area by means of shapes that are placed in such a way as to leave no space between them and to have no overlap of the shapes. See the examples provided (pages 1038 and 1039).

Another way to develop a definition of paving is to use the learning method called concept attainment, and to show students examples and non-examples of paving. See the pamphlet That's a Yes! Concept Attainment, which appears in the collection titled Teaching Strategies Series.

You can display the examples of paving and tiling in the classroom and ask the students to add further examples.

Evaluation: you can ask the students to bring in examples of tessellation. What they bring in can be evaluated according to their understanding of what a tessellation is.

The pattern blocks to be used are the yellow hexagon, the red trapezoid, the green triangle, the orange square, and the blue rhombus.

The size of the worksheet can vary; the bigger the rectangle, the more time will be required to cover the surface.

Demonstrate using the overhead that you start by placing the tile in the middle of the area, tracing its outline, and placing it in another spot, and that you continue to work in this way (from the middle out to the sides) until the whole area has been covered (without leaving any space between the tiles).

Exploration

Is it possible to develop criteria for determining whether a shape can be used to cover the area?

First activity:
Give the students (who are working in pairs or in groups) a large number of two-dimensional shapes and ask them to divide these shapes into two groups according to the following criteria:

• shapes that can be used to cover the area; and,
• shapes that cannot be used to cover the area.

Students can be asked to produce a hypothesis regarding this classification (criteria of classification), to verify it, and to compare the results with their hypotheses.

These results can be shared with the rest of the groups.

You can keep the criteria and the results of this experiment posted on the bulletin board in order to be able to come back to them as the unit proceeds.

The students can work in cooperative groups. With them you can establish objectives for cooperative learning, which can be recorded on a large piece of paper. Some of these objectives might be: to encourage each other; to demonstrate one's dissent without being disagreeable; to take turns doing the work, to focus on the work at hand, to summarize, to engage in mediation, to debate an issue, to listen actively, to stay with your group, to speak in a pleasant tone, to help each other, etc. You ca n consult the booklet titled Opening the Door to Cooperative Learning, in the Learning Strategies Series collection.

You can also assign a role to several people or to each person in the group: a reader, a secretary, a spokesperson, etc. The students can also choose their roles. Other resources for information on cooperative learning are given at the end of the unit.

Here is a sample of the shapes that can be used: circle, oval, a variety of polygons (triangles, quadrilaterals, pentagons, hexagons, octagons, etc.). Irregular polygons should also be presented. The different forms can be numbered to make discussion easier.

This is an appropriate time to begin drawing up a list of mathematical and other terms that are specific to this unit. The list can be posted and added to throughout the unit. The students can refer to it. Making an illustrated dictionary can be an activity they do with their projects.

You can also use shapes to be found among the pattern blocks, attribute blocks, tangrams, etc. Many teacher's guides accompanying textbooks contain shapes that you can photocopy. See the worksheets (pages 1043 and 1044) for shapes that can be photocopied and cut out.

Second activity:

The students should find the data they need to complete a table of information on polygons, the measurement of the interior angles of polygons, etc. The worksheet could be used for this activity. The students could use the shapes fro m the first activity to help them fill out the table.

At the end of the activity, ask each group to analyze the data collected and to develop a written list of criteria to use in determining whether a shape can be used to cover the area.

As a class, discuss each group's results.

Compare these results with the results obtained at the end of the first activity. Each group can make its own comparison before sharing it with the larger group.

Third activity:

Students select one or more shapes and cover the area (completely cover the surface).

Using their imagination, the students draw and/or colour the pattern they have created.

Have students discuss real-life situations and careers where this type of activity is used.

Extension

First activity:
Present to the class a number of works by the artist M.C. Escher and make mention of the fact that he was fascinated by regularity and mathematical structure, by continuity and the infinite, and by the latent conflict in each image. Discuss wit h them the sources of his inspiration and what influenced him.

Discuss M.C. Escher's works with reference to the following categories: initial engravings, evenly filling an area, unlimited spaces, circles and spirals in space, reflections, inversions, polyhedrons, relativity, conflict between the plane and space, and impossible structures.

Evaluation: You can give the students new shapes and ask them whether these shapes can be used to cover the area.

Additional lead:
Can you combine the shapes that will not work with others so as to cover the area?

The students can draw up a list (with examples of polygons) or create a table of the polygons that can be combined to tile a surface.

Additional suggestion:

You can undertake an activity involving guided visualization with the students.

You select one particular shape and ask them to visualize this shape and the area to be filled. Can they visualize another way to cover the surface using the same shape? This activity can be carried out at the beginning of the third activity. For more details about using this strategy, refer to the booklet F.Y.I: For your Imagination, Focused Imaging in the Teaching Strategies Series collection.

Second activity:

Escher used geometrical shapes which he would transform so as to generate other interesting shapes. He would then draw and/or colour his piece. Use the worksheets on pages 1038 and 1039 as examples of shapes that have been transformed.

Show the students how to transform a square using translation and rotation.

The students can then cover the surface of the area using the work of M.C. Escher as a model.

Reflection

Ask the students to write down their personal impressions of the activities in this unit, the relationship that exists between mathematics and visual arts, and such creative strategies as feeling the excitement of discovery, looking for the various possibilities afforded by the representation of an idea, and bringing out the contradictions.

Application

Present a scenario for the whole class or ask the students to choose between the following options:

Scenario 1:
Ask the students to perform transformations starting with shapes other than the square, such as the triangle, the hexagon, etc.

Draw and/or colour them.

Scenario 2:
Use the software program Tessellmania to explore different transformations and ways to cover a surface.

Scenario 3:
Make a kaleidoscopic construction. Find three square mirrors measuring approximately 30 cm per side. Glue them together to form a corner. Cut out shapes that have been traced on wood, plastic, stiff cardboard or some other material. Arrange a composition on the mirror's surface. Draw attention to the fact that the figures are completed by their reflection in the mirror. For example, a semi-circle becomes a circle. Glue the arrangement to the surface with a suitable type of glue.

For further details on how to proceed, refer to the article titled Tessellations that begins on page 30 of The Arithmetic Teacher, March, 1990.

Scenario 4:
Mount a project for a Math Fair. You can plan to invite parents and other classes from the school.

Possible projects:

• describe and illustrate how to transform shapes;
• assemble a collection of linoleum tiles, patio bricks, wallpaper samples, etc.; describe the shapes used and the transformations they have undergone;
• compile tessellation products and explore the careers that use them;
• draw up a chronological summary of Escher's life story;
• write and illustrate a dictionary of the terms that are specific to this unit;
• write about what was learned in the course of this unit;
• prepare a report on another artist's work;
• study the concept of visual paradox as it shows up in the work of various artists, such as, for example, Salvador Dali, André Breton, Max Ernst, René Magritte, Miro, Hans Arp, André Masson. Look at fantasy, dreams, eccentric imagery, optical illusions, and surrealistic perceptions in two- and three-dimensional works (consult The Museum Index. For a Better Understanding of Creators and Modern and Contemporary Movements, National Museum of Modern Art, 1992-93, as well as encyclopedias and dictionaries); and,
• look at the additional strategies and other suggestions for adaptation to find more project ideas.

Other Suggestions for Adapting the Unit

• You can study perimeter and area of the different shapes. What happens to the perimeter and the area when a shape is transformed? (The perimeter changes, but the area remains constant.) You can increase or reduce the number of activities that involve a computer.

• You can use the software programs Tessellmania and Escher-Sketch. (TL)
  • One activity would be to ask the student to use the computer to alter a shape, cover the surface, print out the results and explain, orally or in writing, the kinds of transformations that have taken place (translations, rotations, flips, or their combination).
  • The computer provides a way for students who have problems with motor skills to explore the subject and increase their knowledge of tessellations.
  • For another activity, the student is given a shape that has already been transformed and is asked to interpret this shape in a creative way using the drawing functions of the software program; he/she can add eyes, mouths, etc.
  • For another activity, the student selects a piece by Escher, identifies the original shape and, with the help of the computer program, tries to transform the shape so as to reproduce the artist's design.
  • The students can be asked to create a tessellation that would be suitable for a floor, a T-shirt, wrapping paper, etc.

• Explore kaleidocycles (three-dimensional shapes that can turn). Build a kaleidocycle using tetrahedrons. Each student should have 8 copies of the tetrahedron pattern (see the worksheet ). Ask the students to draw and/or colour th em. The tetrahedrons are folded individually. They are then pasted up one after another to form a circle. The circle can be turned, while the tetrahedrons undergo a rotation.

• Explore platonic solids. They can be identified and constructed. There are 5 platonic solids: the regular tetrahedron, the regular hexahedron, the regular octahedron, the regular dodecahedron, and the regular icosahedron. These solids are polyhedron s in which all the sides are regular congruent polygons. Thus the tetrahedron has 4 sides (congruent triangles); the hexahedron has 6 sides (congruent squares); the octahedron has 8 sides (congruent triangles); the dodecahedron has 12 sides (congruent pentagons); and the icosahedron has 20 sides (congruent triangles).

• Find a number of different paving styles that can be created with rectangles (in proportions of 2:1 and/or 3:1). Talk about what these proportions represent. What other proportions could be used? Ask the students to present examples of pavements.

• What happens when you work with irregular pentagons? (See the worksheet ).Can you pave a surface or create a mosaic with these irregular pentagons? Can you use any irregular pentagon to create a mosaic?

• Escher also worked with ideas of limits and infinity. Find specific pieces that have limits. Find a piece that is dealing with the infinite (the Möbius strip). Why does the Möbius strip represent infinity? Make a Möbius strip yourselves to find out the answer. Here is how to make a Möbius strip: take a band of paper, about 35 cm long by 2-3 cm wide, (adding machine tape works well) by the two ends; turn just one of the ends a half-turn and join the two ends with a piece of adhesive tape. You will then have created a circle with a half-turn in it. Ask the students to find a starting point on the strip and put the tip of their pencil on this point; without lifting their pencil, they then draw a line along the middle of the strip, through its entire length. What happens? After having drawn a line on both sides of the strip, the pencil comes back to the starting point. You can also say that this is a piece of paper that has only one side!

• Find pieces by Escher in which the artist works with two-dimensional and three-dimensional shapes.

• What is the smallest number of colours required to colour an area so that no two adjoining pieces are the same colour? You can begin this investigation by dividing a surface into 2, 4, 6, etc. parts with straight lines.

• Interpret a fantasy theme by creating a symmetrical drawing. Create a fantasy version of a deck of cards. For example, choose the king, the queen, the jack, or the joker. Use a subject of your choice as the initial visual for the portrait (head and torso). Work with an 18" X 24" sheet of paper. Draw the portrait on one half of the sheet. On the other half, make a copy of the drawing with tracing paper and finish it off with acrylics or poster paints.

• Which letters of the alphabet could be used to pave a surface? Can you use your initial, both your initials? Try it (you can distort the letters and transform them a little).

• Ask the students to perform the following exercise: make a contour drawing of their hand or foot. Next, bring in some object in which the students can see themselves, such as a mirror, a Christmas tree ornament, etc. Each student holds the mirror or the ornament in one hand and studies the hand and its reflection. Distortion should be encouraged. Find the works of M.C. Escher in which distortion is present.

• You can explore visual arts from the perspective of distortion and the surreal:
  • Create a surreal landscape inside a shoe-box using a combination of different techniques and two- and three-dimensional materials, such as, for example, photographs, found objects, and shapes that have been sculpted, molded or painted, which you arrange together inside the shoe-box.

  • Construct a Neo-Dadaist collage from various images that have been cut out of magazines, newspapers, photocopies or archival photographs. Assemble them by carefully cutting and pasting them.

  • Compose a bizarre story about an experience or an observation. Write down the story of a real event or experience. Then write a story recounting a dream that seemed especially bizarre or surreal. Illustrate these stories using the medium of your choice (refer to The Book of Images, Éditions Atlas, page 60).

• Create a surrealist bust. Take a styrofoam head and transform it by adding bits of cloth, leather, buttons, string, mechanical gadgets, bottle corks, wool, paint, etc.

• Create a surrealist monument. Make up a surreal event that would rival those in The Book of Records, or use The Book of Records as a resource for ideas, and create a trophy, a medal, or a banner in recognition of this act of braver y.

• Explore the process of colour photocopying as a medium of artistic expression. Colour photocopying is a printing process based on three colours (yellow, blue, and red). The advantage of this process is that you can easily change the desired colo ur. You can also make a black and white print. First make a collage using a selection of images you have clipped from magazines and posters. Arrange the images on a piece of construction paper and glue them down. Add some hand-drawn elements. Use a colour photocopier to make prints of your collage.

• Create an optical illusion by combining graphics, movement, and strobe lighting. Cut a 30 cm circle out of stiff cardboard. Draw a series of concentric circles 2.5 to 5 cm apart. Use acrylics to paint designs within the different segments of the disk. Mount the disk on the transmission of a small motor and attach it to a tabletop. Use a lighting dimmer switch to control the motor's speed. Illuminate the disk with a strobe light. You'll see that the shapes within the concentric circles will show up simultaneously or separately depending upon the motor's speed.

• Perform an integration with other areas of study.
  • Draw up a chronological summary of Escher's life.

  • Locate Escher in history: what other famous people were alive at the same time as Escher, what important events happened in the history of the world or of Canada during Escher's lifetime?

  • Escher had a great deal of trouble with mathematics when he was at school, and yet his works include much mathematical content. How can that be? Ask the students to write about the question. Another famous person who had trouble with mathematics w as Albert Einstein.

  • Then, what is mathematics?

  • Why do you think they call Escher the poet of the impossible?

• Give the students a research assignment to find works of art from different cultures that demonstrate tessellation or symmetry, etc.

• You can study optical illusions.

The program is easy to use and makes use of animation techniques to show students how their pieces change shape and fit together.

Evaluation

The evaluation should be planned at the same time as the unit (what do I want to observe and evaluate? who? how?). Evaluation should reflect the learning activities that have taken place in the classroom. The following are suggestions of activities that can help the teacher to evaluate the student.

• ask the students to bring examples of tiling from home;

• present the students with examples and non-examples of tiling, and ask them to pick out the examples of tiling;

• show the students one or more shapes and ask them to explain why the shapes can or cannot be used to cover a surface;

• ask the students to transform a shape other than a square; and,

• ask the students to write a letter to their parents or to a friend explaining what they have learned during this unit.

These activities can be evaluated by means of a holistic assessment scale or measurement instruments such as those that appear here.

See also Student Evaluation: A Teacher Handbook. (1993). Saskatchewan Education. for information about the holistic rating scales.

Pattern Block Worksheet

Example 1( method of employing a square)

Example 2( method of employing a square)

Different shapes

Different shapes page 2

Activity: Gathering Data

Group members: _______________        ______________

                             ______________     _____________

Tetrahedron Worksheet

Irregular Pentagons Worksheet

Rating Scale

(Discussion involving the whole class or small groups)