Tessellations
Thomas Freese
ArtistinResidence
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 checkerboardstamped pattern on their grid page.
Time Requirement
Two 90minute sessions would allow for the handson 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 twodimensional, 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 samesize 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 miniprints 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" lightcolored construction paper
 craft foam
 doublestick foam
 glue (if doublestick foam is not available)
 21/2" 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
 acute angle: an angle that measures less than 90º
 congruent angles: angles that have the same measure
 equiangular triangle: a triangle with all three angles congruent (of equal measure)
 equilateral triangle: a triangle with all three sides congruent (of equal length)
 glide reflection: a transformation that moves a figure in a slide and mirrors it
 hexagon: a polygon with six sides
 interior (stamp) details: within the exterior line of the shape used for the stamp;
the lines and pattern that create an artistic image
 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
 modified square: the original polygon (square) that has been changed according to
geometric rules in order to tessellate
 mosaic: synonym for tessellation or tiling
 obtuse angle: an angle that measures more than 90º but less than 180º
 octagon: a polygon with eight sides
 paper template: the cutout tracing form of a tessellating shape used to construct
the stamp
 parallelogram: a quadrilateral whose opposite sides are congruent and parallel
 pentagon: a polygon with five sides
 perpendicular lines: lines that meet at right angles in a plane
 plane (surface): a twodimensional, flat surface that is infinite
 polygon: a simple closed shape, bounded by line segments
 print registration: lining up the stamped image according to the grid guidelines and shape
 quadrilateral: a polygon with four sides
 rectangle: a quadrilateral that contains four right angles
 reflection (in a plane): a transformation that mirrors a figure in a plane
 regular polygon: a polygon with all its sides and all its angles congruent
 rhombus: an equilateral quadrilateral
 rotation (in a plane): a transformation that turns a figure about a point in a plane
 scalene triangle: a triangle with sides of three different lengths
 tessellation (plane): a covering of a plane, without any gaps or overlaps, by a pattern of
one or more congruent shapes
 tessellation (space): a filling of space, without any gaps or overlaps, by a pattern of
one or more threedimensional shapes
 tiling: synonym for tessellation or mosaic
 transformation: in this lesson, a movement of a figure to a new location, leaving the
figure unchanged in size and shape
 translation: a transformation involving a slide of a rigid figure without rotation
 translational symmetry: characteristic of a figure that coincides with itself after an
appropriate translation or slide
 trapezoid: a quadrilateral with exactly two parallel sides
 vertex (of a polygon): the point of intersection of any two adjacent sides of the polygon
 vertex (of an angle): the point of intersection of the two rays that form the angle
Lesson Instructions
Guide students through the following process:
 Place your finger on the bottom right corner of an index card or other piece of stiff paper stock.
 Using a ruler to measure the distance, place a pencil mark 11/2" to the left on the bottom
of the card and 11/2" above the right corner of the card, on the right edge.
 Place your ruler upright on the bottom 11/2" mark so the ruler is parallel to the right
edge of the card.
 Using the 11/2" mark on the right edge of the card as a guide, mark a point 11/2" up.
 Using the ruler, connect the three 11/2" marks to form a 11/2" square at the corner
of the index card.
 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.
 Now make two changes to the square, going from its two open sides. For example, you
could draw a simple halfcircle 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.
 Cut out the two pencildrawn changes, taking care to cut along the drawn lines and to remove the
cutouts in one piece.
 Slide the two cutout pieces to their respective opposite sides and place them, pointing the same
direction as the cutout sections, even with the square’s line. Trace the cutout pieces and then
cut out the entire shape for a final, tessellating template or tracing form.
Making a FullPage Grid
 On a piece of lightcolored 9" X 12" construction paper, find the center by marking
a point 41/2" 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.
 Draw a tiny circle around the point where these two lines intersect. This is the center point
of the paper.
 Measure and mark every 11/2" along the two center lines. Then use these marks to draw
horizontal and vertical lines.
Assembling the Stamp
 Place your tracing form on top of the craft foam and trace and cut out the shape.
 Repeat Step 1 with the doublestick foam.
 Check to see whether the two cutout foam pieces are accurate to the tracing form.
 Remove one side of the doublestick foam’s paper covering and mount the craft foam.
 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.
 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
 Always line up the corners of your 11/2" square stamp with the inside corners of the
squares on your 9" X 12" construction paper grid.
 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.
 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, foamcore sheets and suspend them as mobiles in a large indoor space.
 Make Tshirts 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 studentwritten 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
Tshirts; etc.
Additional extension ideas:
 constructing larger stamps for a paperprint mural
 designing tessellating figures on a computer
 making tessellating shapes out of wood for a puzzle
 finding and photodocumenting 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 nontessellating 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) 8721100. The company
offers an extensive catalog that includes these items as well as other K8 educational and
teacher resource materials in mathematics, science, and the arts.
 Symmetry and Tessellations activities links
 Craft foam (the nonsticky variety) is available from local craft stores or from S&S Arts
and Crafts, Norwich Avenue, Colchester, CT 06415, (800) 9373482.
Last Updated: Monday, 29Dec2008 15:23:24 EST
