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Tessellations

Thomas Freese

Artist-in-Residence


In Tessellations, artist/instructor Thomas Freese shows students how they can make a tessellating stamp and use it to create an interlocking pattern. The lesson integrates mathematics and art as the children use geometry, measurement, repetition, and patterning to create unusual, appealing designs.

Lesson Focus

An introduction to the world of tessellations: definition, examples, and the construction of a tessellating stamp. Children create a paper template of a modified square, a grid based on that square, a foam and Plexiglas stamp, and a checkerboard-stamped pattern on their grid page.

Time Requirement

Two 90-minute sessions would allow for the hands-on work and a minimum of discussion on the geometry involved.

Skill Development

Tessellations encourages children to explore the cognitive and observational skills required to understand relative conservation of space in a two-dimensional, repetitive pattern of an interlocking shape or unit. The activity allows children to learn and review basic geometric terms, definitions, and theory, including regular polygons, lines, angles, points, etc. Children employ basic mathematical skills in creating their stamps and their tessellating art: They use a ruler to measure and form a grid with sections and parallel borders, they find the center of the page, and they construct a uniform stamping grid of same-size squares.

The activity requires dexterity and coordination of craft materials to create a stamp within acceptable standards. This is a basic lesson in understanding the printing process: how images reverse and the need to register the mini-prints precisely within the grid guidelines. And it allows the child to create a recognizable creature or symbol from an irregular tessellating contour.

Purpose

This lesson helps students understand tessellations through a combination of art and mathematics concepts and then put their intellectual understanding to work through the construction of a paper template of tessellating shape, the use of measurement to make a stamping grid, and the creation of a stamp with a unique and possibly recognizable image.

Sources for Examples of Tessellations

Artists
  • M.C. Escher
  • Bridget Riley
  • Victor Vasarely

Mathematicians

  • Johannes Kepler
  • Archimedes

General

  • tile and/or construction patterns
  • geometric quilt patterns
  • soccer balls

Connections to Educational Standards

The following Kentucky Academic Expectations are all related to Tessellations:

  • 2.9: Students understand space and dimensionality concepts and use them appropriately and accurately.
  • 2.10: Students understand measurement concepts and use measurements appropriately and accurately.
  • 5.1: Students use critical thinking skills such as analyzing, etc.
  • 5.2: Students use creative thinking skills to develop or invent novel, constructive ideas or products.

Materials Needed

  • index cards
  • pencils
  • rulers and scissors
  • 9" X 12" light-colored construction paper
  • craft foam
  • doublestick foam
  • glue (if doublestick foam is not available)
  • 2-1/2" Plexiglas squares
  • stamp pads
  • ink
  • scrap paper to protect surfaces from ink

Alternative Materials/Sources for Materials

  • index cards: any card stock that is crisp, not soft like construction paper. Tag board or scrap paper from a print shop is fine.
  • Plexiglas: Plexiglas scraps are available from hardware, glass, or frame shops. They are cheap; in fact, store owners often will donate them to teachers or schools. Other materials, such as cardboard, could be used to mount the stamps; the advantage of Plexiglas is that you can see through it.
  • Dale Seymour (see Resources below) makes materials that stamp: a foam sponge class kit for printing on paper and fabric. Students could also carve potatoes, erasers, or linoleum blocks to create stamps for this activity.
  • doublestick foam: If doublestick foam is not immediately available, you can glue together two layers of craft foam to make the stamps. Craft foam is available from arts and crafts stores. If you don’t have an arts and crafts store in your community, look for craft foam and/or doublestick foam at office supply stores.
  • If your students carve rubber into stamps, you won’t need Plexiglas, craft foam, or doublestick foam at all; but this method is recommended for older students only (4th grade and up).

Vocabulary Related to the Lesson

  1. acute angle: an angle that measures less than 90º
  2. congruent angles: angles that have the same measure
  3. equiangular triangle: a triangle with all three angles congruent (of equal measure)
  4. equilateral triangle: a triangle with all three sides congruent (of equal length)
  5. glide reflection: a transformation that moves a figure in a slide and mirrors it
  6. hexagon: a polygon with six sides
  7. interior (stamp) details: within the exterior line of the shape used for the stamp; the lines and pattern that create an artistic image
  8. line of reflection: a line in a plane that lies equidistant from any two corresponding opposite points in a figure that has reflective symmetry; also called mirror line
  9. modified square: the original polygon (square) that has been changed according to geometric rules in order to tessellate
  10. mosaic: synonym for tessellation or tiling
  11. obtuse angle: an angle that measures more than 90º but less than 180º
  12. octagon: a polygon with eight sides
  13. paper template: the cut-out tracing form of a tessellating shape used to construct the stamp
  14. parallelogram: a quadrilateral whose opposite sides are congruent and parallel
  15. pentagon: a polygon with five sides
  16. perpendicular lines: lines that meet at right angles in a plane
  17. plane (surface): a two-dimensional, flat surface that is infinite
  18. polygon: a simple closed shape, bounded by line segments
  19. print registration: lining up the stamped image according to the grid guidelines and shape
  20. quadrilateral: a polygon with four sides
  21. rectangle: a quadrilateral that contains four right angles
  22. reflection (in a plane): a transformation that mirrors a figure in a plane
  23. regular polygon: a polygon with all its sides and all its angles congruent
  24. rhombus: an equilateral quadrilateral
  25. rotation (in a plane): a transformation that turns a figure about a point in a plane
  26. scalene triangle: a triangle with sides of three different lengths
  27. tessellation (plane): a covering of a plane, without any gaps or overlaps, by a pattern of one or more congruent shapes
  28. tessellation (space): a filling of space, without any gaps or overlaps, by a pattern of one or more three-dimensional shapes
  29. tiling: synonym for tessellation or mosaic
  30. transformation: in this lesson, a movement of a figure to a new location, leaving the figure unchanged in size and shape
  31. translation: a transformation involving a slide of a rigid figure without rotation
  32. translational symmetry: characteristic of a figure that coincides with itself after an appropriate translation or slide
  33. trapezoid: a quadrilateral with exactly two parallel sides
  34. vertex (of a polygon): the point of intersection of any two adjacent sides of the polygon
  35. vertex (of an angle): the point of intersection of the two rays that form the angle

Lesson Instructions

Guide students through the following process:

  1. Place your finger on the bottom right corner of an index card or other piece of stiff paper stock.
  2. Using a ruler to measure the distance, place a pencil mark 1-1/2" to the left on the bottom of the card and 1-1/2" above the right corner of the card, on the right edge.
  3. Place your ruler upright on the bottom 1-1/2" mark so the ruler is parallel to the right edge of the card.
  4. Using the 1-1/2" mark on the right edge of the card as a guide, mark a point 1-1/2" up.
  5. Using the ruler, connect the three 1-1/2" marks to form a 1-1/2" square at the corner of the index card.
  6. Along each side of the square, measure off 1/2" spaces, making two marks evenly spaced along the side. Carefully connect the marks to form two sets of parallel lines—two horizontal lines and two vertical lines.
  7. Now make two changes to the square, going from its two open sides. For example, you could draw a simple half-circle going in from the bottom side and a triangle going in from the right side.
    Hints: Avoid placing changes on the corner. Also, simpler figures will be easier to cut out with your scissors.
  8. Cut out the two pencil-drawn changes, taking care to cut along the drawn lines and to remove the cut-outs in one piece.
  9. Slide the two cut-out pieces to their respective opposite sides and place them, pointing the same direction as the cut-out sections, even with the square’s line. Trace the cut-out pieces and then cut out the entire shape for a final, tessellating template or tracing form.

Making a Full-Page Grid

  1. On a piece of light-colored 9" X 12" construction paper, find the center by marking a point 4-1/2" up on each of the 9" sides. Then connect these two points with a straight line. Then place a mark at the 6" center point of the 12" sides and connect these two points with a straight line.
  2. Draw a tiny circle around the point where these two lines intersect. This is the center point of the paper.
  3. Measure and mark every 1-1/2" along the two center lines. Then use these marks to draw horizontal and vertical lines.

Assembling the Stamp

  1. Place your tracing form on top of the craft foam and trace and cut out the shape.
  2. Repeat Step 1 with the doublestick foam.
  3. Check to see whether the two cut-out foam pieces are accurate to the tracing form.
  4. Remove one side of the doublestick foam’s paper covering and mount the craft foam.
  5. Remove the other side of the doublestick foam’s paper covering and attach the combined forms to the Plexiglas. The Plexiglas provides a mount for your stamp.
  6. If you wish, use a pencil to press into the foam and draw line patterns. These will stamp out white and provide interior details to the tessellating shape.

The Stamping Process

  1. Always line up the corners of your 1-1/2" square stamp with the inside corners of the squares on your 9" X 12" construction paper grid.
  2. Working with a partner and using a single color of ink, fully stamp the page in an alternating pattern to create a checkerboard effect (half of the squares will be stamped and half blank). Place scrap paper under your grid to protect your desk or table from ink stains. Keep your stamp in the same position; don’t flip or rotate it.
  3. Rinse your stamp in running water, dry it, and then start stamping the blank squares in a second color. Once again, keep your stamp in the same position, neither flipping nor rotating it.

Response to Art

Children can create creatures, designs, or messages within the form of their tessellating shape. They can talk about these embellishments and how they fit in the contour of the shape. They also can review and report on the basics of the lesson: the definition of tessellation, examples, how they created a modified parent polygon (from a square), how they made a grid and stamp, what challenges to stamping or printing they encountered, and how they solved these problems (they may have learned to inscribe letters in reverse into the foam).

Exhibition Suggestion

  • Make cards for a sale.
  • Stamp on large, foam-core sheets and suspend them as mobiles in a large indoor space.
  • Make T-shirts to wear.
  • Put the 9" X 12" stamped paper sheets all together in a hallway mosaic.
  • Laminate the prints and use them as placemats.
  • Put prints on posters (along with student-written instructions) for display and for teaching other students about tessellations.

Extensions

What would the students like to do next with tessellations? Possibilities include more stamping; creating a different stamp, different details, or a different parent polygon and grid; stamping T-shirts; etc.

Additional extension ideas:

  • constructing larger stamps for a paper-print mural
  • designing tessellating figures on a computer
  • making tessellating shapes out of wood for a puzzle
  • finding and photo-documenting tessellations in building construction
  • researching Islamic tile patterns
  • doing a study of M.C. Escher and the influences that led to his work with tessellations
  • doing a video interview with a quilt maker
  • creating new kinds of tessellations by starting with different parent polygons (rectangles, triangles, hexagons)
  • researching the geometry—the sum of the angle measurements—which proves the tiling theory
  • creating prints on fabrics
  • creating tessellating shapes from fabric and sewing them together
  • locating a tessellation in your home or town, sketching it, and writing a basic analysis
  • studying patterning in nature (Have a beekeeper visit and show the hexagons of the honeycomb.)
  • cutting out non-tessellating polygons and experimenting to find repeating shapes that could fill in the gaps
  • exploring M.C. Escher’s books to discover and analyze underlying grids Students’ papers about tessellations, along with examples of the tessellating patterns they have created, make excellent portfolio entries.

Resources

  • books by and about the Dutch artist M.C. Escher
  • local quilt makers
  • mathematics teachers
  • Books, materials, manipulatives, and posters related to tessellations are available from Dale Seymour Publications, P.O. Box 10888, Palo Alto, CA 95303, (800) 872-1100. The company offers an extensive catalog that includes these items as well as other K-8 educational and teacher resource materials in mathematics, science, and the arts.
  • Symmetry and Tessellations activities links
  • Craft foam (the non-sticky variety) is available from local craft stores or from S&S Arts and Crafts, Norwich Avenue, Colchester, CT 06415, (800) 937-3482.



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Last Updated: Monday, 29-Dec-2008 15:23:24 EST