This unit integrates math with language arts and arts
education. Throughout the activities in this unit, the students are asked to respond to
the concept of tessellations. They will write definitions and assess how well
their group worked together. In addition, we have exposed the students to
tessellations in the environment, such as interior design. If one can locate
some prints by M.C. Escher, students will see how their topic can become a
famous art form. As a culminating activity, students will create a classroom
quilt. The quilt will be a combination of a geometric shape (the square piece
of paper) and a geometric transformation (translation, rotation, or
This mathematical topic can also be explored in other subjects. Quilts and
their importance can be studied through social studies. The
classroom quilt can be considered symbolic for the class; each student could
compose a story to explain her/his quilt patch. In science, when students are
studying insects, they can connect tessellations to the wings of a
fly or butterfly. These are some of the ways we think our topic of
tessellations can be integrated across the curriculum.
Communication - students will communicate their understanding of the
tessellation concepts in written form. Since a number of lessons involve the
teacher modelling a technique, students will have an opportunity to develop
their listening skills. Because there is a lot of direct teaching involved, it
will require the teacher to use clear and precise language. Students will also
be encouraged to use "math language" - rotation, slide, trapezoid, hexagon,
Numeracy - Students will engage in problem solving as they discover the
different ways one can tessellate polygons from the geoblocks, and how to
tessellate a shape using slide, rotation, and reflection techniques. During the
introductory lessons, students will discover the characteristic traits of
polygons and why certain shapes in and of themselves will not tessellate.
Students will also predict certain outcomes regarding whether or not a shape
Personal Social Values and Skills - Students will work cooperatively and
contribute positively in the group learning activities.
Independent learning - Students will be
discovering tessellation patterns, which will lead to independent
Because our classroom is already arranged into groups of four at tables, we decided
to use this set up for our cooperative learning lesson. Most of the lessons in
this unit involve individual work, although pairing and grouping could be used
as an adaptation to students' needs.
Materials used in this
unit are mainly geoblocks, paper, scissors, pencil crayons, and tape. The
geoblocks are normally stored in containers in or on cupboards in the
classroom. Recycled paper is recommended, when appropriate, so this paper
would come out of the classroom recycling bin. Students should have their own
pencil crayons and scissors. The teacher will need to provide tape.
Portfolios will be used to organize student work. Checklists and rating scales
with set criteria will be used to check the students' understanding of tessellations
and the procedure of tessellating. While students are working, anecdotal notes
will be taken regarding the progress of each student and their involvement in each
The teacher reflections should be taken throughout the unit, noting areas
that need more work and if unit objectives were met. Reflection can also
generate new ideas, and help the teacher to decide on necessary changes or
adaptations that may be needed.
Students will have an opportunity to evaluate their own progress and
understanding of the unit content. Students will be provided with a rating
scale and checklist and asked to give themselves a mark for how well they understood the
concepts related to tessellations.
When working in groups, students will have the opportunity to evaluate
themselves as a group using rating scales and check lists.
Set / Development: In a class discussion, the students will be
asked what they notice
about the shapes, particularly commonalities and differences. The students will be told
that these shapes are called polygons (definition - a closed plane figure
bounded by straight lines).
At this time the students will be introduced to the concept of tiling.
Polygons can be used to tile a surface. When tiling there can be no gaps or
overlaps. To demonstrate what tiling is, the square blocks from the geoblocks
will be used on the overhead projector.
Then the students will be asked to predict which shapes will tile
|parallelogram / rhombus
The students will be reminded to look at the square for the example.
- We will discuss/review the concept of predicting (a prediction does not have to be correct, it is just a guess)
An example / non-example chart will be used. Each shape will be voted on by the students as to
whether it is an example or nonexample of tiling. Each table will use a set of shapes to see if they will tile a plane.
Students will work individually for this activity. After the students have had time to decide whether the shape can tile we will
regroup, revisit, and revise the example/nonexample chart based on the information the
Closure: Students will write a definition of polygon and what
they learned about them in their journals.
Extension: To prepare the students for the next class activity, we will discuss
cooperative learning. Some goals for the groups may include letting
everyone speak, taking turns talking, and to speak kind and encouraging words
rather than put downs.
Set: For the first 5 minutes we will review what we learned in the previous class. We
will discuss the definition of a polygon at which time the students will have the
opportunity to revise or make additions to the definition they wrote in their
journals. We will review what it means to work cooperatively (working together, taking
turns, being polite, etc.).
Development: Introduce the term tessellation and relate to what
- Each table receives one set of shapes and a piece of chart paper.
- Each student will use a different colour pencil crayon and working together
they will tessellate the polygon on the chart paper.
- The students will write their names on the paper in the colour they
used or create a key.
| TABLE 1
| TABLE 2
| TABLE 3
| TABLE 4
| TABLE 5
| TABLE 6
| TABLE 7
As they finish tessellating, the group raises their hands to indicate they are
One of us will be available to check on the group, answer questions etc.
- Each table will receive 2 or more shapes.
- Using the same process as above (tracing with different coloured pencil
crayons) the students will tessellate the other side of the chart
| TABLE 1
|| Triangle, Hexagon
| TABLE 2
|| Hexagon, Square, Triangle
| TABLE 3
|| Octagon, Square
| TABLE 4
|| Dodecagon, Triangle
| TABLE 5
|| Dodecagon, Square, Hexagon
| TABLE 6
|| Square, Triangle
| TABLE 7
|| Trapezoid, Triangle, Kite
Closure: Students will regroup and share with the class what they
Students will report in their journals about how well their group worked together.
Objective: to recognize tessellations in the environment
- Return math journals to the students for an opportunity to add and/or
revise their definition of tessellating (tiling). Discuss as a class.
- Explain that there are tessellations everywhere in the environment.
- Discuss environment (any place outside one's personal space, like the classroom, for example).
- Sitting in their desks, students will look around the classroom in
search of tessellations. Ask each student for an example.
The class will be divided in half. One half of the class will go on a walking tour
of the school in search of tessellations. The other half of the students will look through provided books in search of tessellations in the environment. After 15-20 minutes
the groups will switch.
Closure: Ask the students to share any tessellations they really
them to write five examples they observed, either in the
books or around the school, in their journal.
Self Evaluation from experience in the field:
- needed more examples before asking for examples from students.
- divide "problem people" up right away. Make them sit out if the
- concept chart for tessellation/non-tessellation.
Objective: Develop visual and conceptual understanding of translation
- Give each student a piece of paper that has been tessellated using a
slide. One of the shapes will have the corners numbered 1 to 4.
- Give each student a single copy of the tessellated shape so that they may
manipulate it. It will also have the corners numbered the same way as the
- Explain that there are two different ways to tessellate:
- the way we have been doing it, using polygons.
- and the way we will study for the next while - transforming the
- Using the overhead, demonstrate the activity using a different
- Place a single shape over the tessellated shape that has the
corresponding numbers on it.
- Slide the shape to the one beside the original and write the numbers in
the correct corners. Continue for two more shapes.
- Ask the students what they have observed. Ask them to describe the demonstrated procedure:
- line up the edges,
- lift the shape, and
- write down the corresponding numbers.
- Students will predict what will happen when they try it.
- Now the students will repeat the procedure with their sheet of paper (10
- Were the predictions confirmed? Do they see any patterns on their
Explain that this way of tessellating is called the slide translation. Why do
they think this is so? Discuss in small groups and then tell students to
decide on a definition and write it in their journals.
Extension: Flip paper over and begin to tessellate it using the same
Evaluation: Students will hand in their sheets and their journals.
list will be used to note if they understood and performed correctly the
procedure and if they wrote a definition for slide translation. The definition
does not have to be correct in order to receive credit.
SLIDE TRANSLATION: Tessellating a shape
across a surface, without flipping it or changing the position of the
Objective: Creating shapes that tessellate and developing the nibbling technique for geometric transformations.
- Students will receive a blank sheet of paper and the same shape from
Activity 4. They will be encouraged to tessellate using the slide (5 minutes).
- Review and revise their definitions for slide translation.
- Tell students that we will be exploring a way to make our own
shapes tessellate. We will use the nibble technique to change an ordinary
square into a different shape.
- Hand out 3 x 3 recipe cards, one for each student. Tell them to
one side of the card (preferably the one with lines on it).
Using a series of overheads, demonstrate the nibbling
- Draw a simple design from one corner of the sheet to an adjacent corner.
(Do not draw diagonally) Stress to the students that they must draw from corner to corner. Do not
stop halfway across!
- Cut on the design line, being sure to have 2 pieces when done -the nibble
and the rest of the sheet. There should be no other pieces laying around.
This is very important! No trimming allowed.
- Slide the nibble across the sheet to the opposite side and tape the
straight edges together. Do not attach it to an adjacent edge. Do not flip
the nibble around so that the coloured side is showing. Do not overlap the
edges when taping. The corners of the piece and the nibble should match
- Repeat the procedure for the other sides, but ask the students to tell you
how to do it.
- Encourage students to make their own slide tiles using the nibbling
For those needing more guidance, have them come to the front of the room with
their cards and go over the procedure with them individually. Use an oversized version of
their card to demonstrate, if needed.
- Encourage them to use simple designs for their first attempt at the nibbling
technique. This will hopefully lead to success at all levels.
- Students will use their slides to tessellate an 8 1/2 x 11 sheet of
time allows, they may colour their pattern.
Evaluation: Paperclip the slide and 8 1/2 x 11 sheet together and collect them.
Assess the quality of the nibble and the tessellated paper.
Objectives: To expose students to another form of tessellations which is the result of rotating the polygon. To help students understand that the reason some
shapes tile is because the sum of the angles around any point will equal
360o. If it does not, the shapes will not tessellate.
- Students will be divided into six groups to tessellate a polygon at each
of the stations. They will be asked to think about what they had to do to the
polygon in order to get it to tessellate. (i.e. how did they manipulate it).
- Then they will be instructed to number each of the corners and trace a
polygons using rotation and record the number in the corners.
- Students will be asked to explain what they observed and if they saw
any patterns within the numbers.
- It is intended that this unit would follow after a unit of measurement -
angles. Tthe student would have a general idea of what angles are. The
student will be given the degrees for each angle of their polygon.
- Ask the students to trace their polygon. Have the students add the sum of
each of the
polygons. They will be asked to write the degree of the angles in the corresponding corners and
then rotate their polygon and add up the angles around the vertex.
- Learning Environment
- The class will be set up with six stations with the
following polygons (pattern blocks) at each station - square, triangle, rhombus (beige), rhombus
(blue), trapezoid, and hexagon. For the set,
the student will be asked to rotate from station to station and do the same
- Students will be asked to write a definition of rotation in their
math journal. These will be checked for accuracy. Since students will be
asked to trace their polygon using rotation this will also be used to determine
if the student understands rotation.
- Rotation is restricted to polygons with adjacent sides that are
Objectives - This is an extension of rotating using the nibbling
Students will realize that they can alter a polygon's shape and still have it
- We will review the term rotation.
be shown examples of tessellations with rotation and the use of the nibbling technique.
- They will be asked to think about and identify what was done in order
for the shape
to tessellate. As a class we will discuss our ideas.
Evaluation of the rotation concept will be based on whether or not the
student's shape was able to tessellate.
Objectives: To recognize the third type of tessellation (reflection), describe
it, and make a definition for it.
- Explain to students that they will be getting a sheet of paper with
- Explain that only one triangle's points are numbered.
Hold up a large example of a reflection tessellation with only one shapes'
- Show students that they are going to have to number each
point of every triangle so that the numbers correspond to the points on the
- Explain that once the points have been numbered, the
students are to decide whether the triangle has been" flipped" from its
original position. If it has been "flipped," they
are to colour the triangle. If it has not, they must leave it white.
- They must do this for every triangle (i.e. number points, and colour
- Explain that each student will receive a tessellated sheet, but
they may work in pairs.
- Ask students to review the steps.
- Hand out the tessellated sheets to each student. Explain that
when they have finished numbering the points and colouring the "flipped"
triangles on the entire sheet, students must write down what
they noticed, and describe how they numbered and coloured the triangles, in their journals.
- Ask the students to use correct terminology (slide, rotation, etc.). Allow
students 10 - 20 minutes to work on the tessellations and 5 - 10 minutes to reflect.
- When students are finished, have each pair present their tessellations to the class,
describing what they did (did they rotate and flip? or slide and flip?).
- When each student has finished presenting, ask students to describe their type of
- Ask what happened to the original triangle. Probe if needed.
- Write their answers on the board. Give them the definition of a reflection tessellation. Write it on the board.
- Compare the two definitions. Ask students if they were close to the actual definition.
- Have students write the definition in their journals. Have students
hand in their sheets and journals.
- Checklist to note if students understood and performed procedure
- Check to see if a definition for reflection tessellation has been
Definition of Reflection: tessellating a shape with a combination of
rotating and flipping.
- recall the definition of a reflection tessellation and
- Use the nibble technique to make reflection tessellations
- Review nibble technique. Remind students that only two pieces must be
cut. Explain to students that they will be making reflection tessellations.
- Show students on a large piece of paper how to draw a design, point to
cut it out, then slide one piece to the opposite side and flip it over. Tape
it into position.
- Do this on two sides. Show how it tessellates (trace it a few times,
sliding and flipping, rotating and flipping).
- Review the steps for nibbling a reflection tessellation. Ask if there
are any questions.
- Hand out a piece of white paper, and a square card to each student.
students that when they are done they are to write the steps they used for tessellating in their journals.
- When they have finished tessellating and writing in their journals they
can colour their tessellations.
- Allow 20 minutes for students to make their shapes and
to tessellate them.
- Have each student present their tessellations. Have other students observe
each other and ask the students observing to tell whether it fits the criteria of
reflection tessellation (has it been flipped, slid, or rotated?).
- Ask students to think about and name the three kinds of tessellations
Ask students to think about what kind of tessellation they like the best and
what kind of design they would like to do for a quilt.
Evaluation: Have students hand in their tessellation paper and their
shapes. Have a
checklist with criteria: creativity, precision of tessellation, and
- Ask students to think about and name the three kinds of tessellations
- Ask students to think about what kind of tessellation they like the
best and what kind of design they would like to do for a quilt.
Development: Using a quilt as an example, discuss how the class can
quilt using the three types of transformations.
- they may choose to use one of their cards from the other lessons,
may make a new one.
- Give them an 8 1/2 x 8 1/2 piece of paper or a 3 x 3 recipe card.
- Encourage them to challenge themselves with a harder transformation.
Evaluation: Assess the quality of the "quilt block":
- Do the shapes tessellate well?
- Did the student take the time to do a good job?
- Is it neat? Colourful?
- Is the shape creative?
- Tesselation Resources
- provides references about tessellations and sources for illustrations
(approx. 45 resources)
What is a Tesselation
- answers what is a tessellation
- opportunity to ask Dr.Math about tessellations
- this site provides designs that students have created
- investigating tessellations using pattern blocks
- links to activities using pattern blocks in colour
- tessellation links 15-20 sites
Bolster, Carey. Activities: Tessellations. Mathematics Teacher 66(April 1973): 339-342.
Clauss, Judith. Pentagonal Tessellations. Arithmetic Teacher 38(January 1991): 52-56.
Giganti, Paul and Mary Jo Cittadino. The Art of Tessellation. Arithmetic Teacher 38 (March 1990): 6-16.
Kaiser, Barbara. Explorations with Tessellating Polygons. Arithmetic Teacher 36 (December 1988): 19-24.
Maletsky, Evan. Designs with Tessellations. Mathematic Teacher 67 (April 1974): 335-338.
Van De Walle, Jon. Let_ Do It: Concepts, Art and Fun From Simple Tiling Patterns. Arithmetic Teacher 28
Tessellations Within Nature
Hiebert, C.E. (1995). Gift of Wings. Erin, Ontario: The
Muir, R. (1986). Landscape and Nature Photography. London: George
Nuridsany, C. and Perennou, M. (1976). Photographing Nature. New York:
Oxford University Press Inc.
Wignall, J. (1987). Landscape Photography - A Kodak Guide. Rochester,
NY: Photographic Products Group.
Tessellations Within Craft
Better Homes and Gardens (1990). Christmas Quilts and
Duff-Williamson, D. (1992). Sensational Scrap Quilts. Paducah, KY:
American Quilter's Society.
Perry, G. (1995). Impressionist Quilts. Lafayette, California: C
& T Publishing.
Tessellations Within Interior Design
Diamonstein, B. (1982). Interior Design - The New Freedom. USA:
Rizzoli International Publications, Inc.
Marberry, S.O. and Zabon, L. (1995). The Power of Color. Toronto:
John Wiley & Sons, Inc.
Pile, J.F. (1988). Interior Design. New York: Harry N.
Artistic Landscaping Supplies (1996) Ltd. 5901 13th Avenue, Regina.
Evaluation / Assessment Tool
Seymour, D. (1982). Tessellation Teaching Masters. Canada: Dale Seymour
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