Internet Lesson Plans

by Woody Duncan

            <<<<contact me at mailto:wduncan@kc.rr.com      my web site is http://www.taospaint.com >>>>
 
 

TESSELATION LESSON PLAN
for 7th Grade Wheel Art
at Rosedale Middle School

 
The word tessellation comes from the Latin Tessella, which was a small Square stone or tile used in ancient Roman mosaics.  Tiles and Mosaics are common synoyms for tesselations.  A plane tessellation is a pattern made up of one or more shapes, completely covering a surface without any gaps or overlaps.
Some shapes, or polygons, will tessellate and others will not.  As for the regular polygons, tessellations can easily be created using squares, equilateral triangles and hexagons.  I choose to use the equilateral triangle because of the unique visual arrangements it seems to lend itself to.  It seems to both perplex and then amaze first the creator and then the viewer.
FOCUS - The major focus of this lesson is to surreptitiously introduce the teaching of math concepts across the curriculum.
OBJECTIVE / OUTCOME - The student will produce a tesselation of original design, developed using an equilateral triangle, where three unique symmetrical faces rotate through out their creation.
MOTIVATION / EXEMPLARS -The art of M C Escher and the mosaic designs of thr early Moorish Mosques are excellent examples that can serve to motivate students and serve as historical exemplars.
VOCABULARY  FOR  TESSELLATIONS
TESSELLATION PLANE (SURFACE) EQUILATERAL TRIANGLE
TWO-DIMENSIONAL RHOMBUS TILES
SYMMETRY TRANSFORMATION REFLECTION
COMPASS HEXAGON CENTER OF ROTATION
POLYGON MIRROR IMAGE ROTATION
MOSAIC REPETITION VARIETY

PROCESS / METHODS

We begin by drawing symmetrical faces.  I give each student an 8 1/2 x 11 with 4 vertical lines.  They are to draw half of a face (front view) next to each vertical line.  Then they fold on the lines and carefully trace the other side of each face.  Unfolding the paper should produce a complete "symmetrical" face, each side being a reflection of the other.  Even at this point I instruct students to make each face as different as possible from any of the others.  This initial step is an opportunity to develop ideas, and to practice the skill of tracing.  We find the classroom window to be better for this purpose than using a light box.
Worksheets with two equilateral triangles (rhombus) should be made available to the students next.  Some may start over several times.  The teacher needs to insure that the triangles they are given are as perfect as possible.  Each minor error increases the problem of correctly matching the finished designs into a tesselation.
A DEMONSTRATION
BY THE TEACHER
IS NECESSARY 
FOR SUCCESS
Students should be instructed to draw three 1/2 faces on the inside edges of "one" triangle only.  The faces should be as different from each other as possible, to provide variety in the design.  I often use as examples - a clown. a cowboy & a little girl (or) an angel, the devil & a monster to begin them thinking about contrast.  The final designs rotate best if each of the three faces points in a different direction.  In other words, if no two tops of heads point toward each other.
Each of the three faces should somehow fill the entire space available.  This is best done in a creative manner, for example - the hair on one head can become the fur collar on the next face when viewed from a different perspective.  Another two faces might share lines and shapes that viewed one way looks like the crown of a queen, but from another angle becomes the shirit of a sailor.  The most difficult problem is utilizing the space in the center, of the triangle, between the three heads.  I should not be left empty.  It works best when somehow the shapes of the three 1/2 faces blend together to fill this space.
After the three faces inside one triangle are finished, the lines should be darkened.  (This makes them easier to trace and they will copy much better, later on the copy machine).  Next carefully fold the paper so that the two triangles line up percisely as seen through the folded paper.  Explain to the students that they are folding the rhombus on the center line and that the edges of the paper itself will, most likely, not line up.  At this point the students should carefully trace the image of the first triangle onto the second.  This traced reflection should be an exact copy (as much as possible) inorder for later xeroxed copies to match up with each other.  A more dramatic and visually striking design will be produced if the students also darken in areas in the design.  This too must be symmetrical.
If you Have access to a good xerox machine, twelve copies should be made of each design.  If the students put their name outside the triangles, their name will be on every copy made.  When cutting out the 12 rhombus shapes, do it carefully and exact, one at a time.  They must be as percise as possible.

Practice laying out the design before trying to glue it down.  Arrange 6 shapes into a six pointed star first.  Then the remaining shapes should fit in between the points of the star to create a hexagon.  Of course if you made more copies and had a larger surface to cover, you could have your design continue on and on.

Glue shapes down carefully, making sure they match.  You may need to adjust shapes or overlap a bit to make everything fit.  Each error in drawing, folding, tracing, copying, and cutting multiplies the problems in getting the final design sections to aline.

PROBLEMS -The image (Rhombus/Triangles) sometimes seem to get streached while they are being copied in the machine.  The teacher needs to be aware of this and watch for the problem.  Make the students aware also so they do not become frustrated with their own abilities.  In a perfect geometric world, it should work.

ADDING COLOR - Colored markers make these tesselations really look fantastic. To do the color correctly (tessellate the colors also) each time a shape repeats in the design, the color should repeat as well.

MATERIALS NEEDED -
Colored Markers
Practice Worksheets (just for symmetry)
Worksheets with Rhombus
Windows or Lightbox
Compass 
   (to demonstrate making an equilateral triangle)
Straight Edge
Scissors
Glue (Elmers)
White Poster Board (make sure it will fit)

REFERENCES -

The World of M C Escher
M C Escher & J L Locher, 1971

Harry Abrams, Inc,  New York

Introduction to Tessellations
Dale Seymore & Jill Britton, 1989
Dale Seymore Publications, Palo Alto
 

ONGOING  ASSESSMENT -
Assessment of student learning should be an ongoing monitoring process done while assisting students in the new learning.  Monitor how well students grasp the new concepts and assist only as needed as they progress through each step.  Help enough to avoid frustration but don't solve every problem for them.

FINAL  ASSESSMENT -
Final assessment is done by their peers and other teachers in the building.  When the finished tesselations are put on display, they will be viewed as a very difficult and complex endevor, at least to those who do not understand how they were created, Let them marvel.  The students who created the designs can keep quiet as to how easy the design was to create, but only if they choose to do so.
 

INFORMATION ON THIS AND OTHER LESSONS -
Can be obtained at http://www.taospaint.com

END