Tessellations
NCTM Principles and Standards for School Mathematics
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Identifies, compares and analyzes attributes of two-dimensional shapes
Investigates, describes and reasons about the results of combining and
transforming shapes
Recognizes geometric ideas and relationships and applies them to other disciplines
and to problems that arise in everyday life
Grade Level: 3-5 (Ages 8-11)
Teacher Instructions
Tessellations can be found in nature, art and in the home and classroom. In this Kidspiration®
activity, students will use Kidspiration Pattern Blocks ™ to develop an understanding of
tessellations and characteristics of shapes that tessellate, review the names of polygons and
explore the relationship between rotation and shape composition. The activity includes single
shape tessellations and multiple shape tessellations, making it easily adaptable for different
age groups and ability levels.
1. Introduce the concept of a tessellation to students. A
tessellation, or a tiling, is a collection of shapes that fit together
without gaps or overlaps to tile a surface or plane. On the
board, or in a new Kidspiration document with pattern blocks,
show how squares can fit together without gaps or overlaps to
tile a surface. Emphasize that if the board or workspace were
larger, the tessellation could extend infinitely.
2. Ask students if they think that circles or ovals could tessellate a plane. Encourage
discussion about why not, pushing students to realize that in order for shapes to
tessellate the plane, they need corners or straight edges. Remind students of the definition
of a polygon, and refine the definition of a tessellation to be a collection of polygons with
no gaps or overlaps.
3. Open Tessellations.kia from Kidspiration Starter>Activities>Math. Read the
instructions aloud, explaining to students that they will be using pattern blocks to create
tessellations. Review the names of all of the polygons in the Math palette. Emphasize
that they must cover the yellow area completely with shapes, without gaps or overlaps.
Remind students that it is okay if shapes extend beyond the edge of the yellow square,
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Tessellations
and that they may need to use the Rotate button on the bottom toolbar to fit the shapes
together.
4. It may be beneficial to begin the first tessellation as a class
so that students understand activity expectations and use of
the Rotate button. Emphasize that it does not matter where
the triangles are placed or how many are used, as long as the
entire yellow surface is covered.
5. Assign students to complete pages 1-5 of the activity in
small groups or independently, depending on computer
availability. While students work, ask them to use the definition of a tessellation to
decide if their designs are tessellations. Encourage a variety of arrangements, noting
that there may be more than one way to arrange shapes so that they tessellate the plane.
linear pattern
flower pattern
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Tessellations
Pages 4 and 5 of the activity involve tessellations with 2 or
more shapes. These are more difficult and depending on the
age and ability level of students, the activity can end after the
completion of page 3.
For students who continue working, there are an increasing
variety of arrangements, and some students may create
regular tessellations (tilings that cover the plane in regularly
repeating patterns) while others may simply arrange shapes in
a non-repeating pattern to cover the surface. See Tessellations Exemplar.kid
for a sample completed activity.
6. Encourage students to rotate around the room so that they have a chance to
view their classmates’ tessellations. Alternatively, pick several student examples
to show on the projector, asking students to note differences in the number and
arrangement of shapes.
7. Lead a discussion while showing samples of student work:
For single-shape tessellations (pages 1-3)
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Did any shape(s) cover the yellow area exactly?
Did it take more triangles or more squares to cover the yellow area on page 1? Why?
Did it take more hexagons or more trapezoids to cover the yellow area on page 2?
Did it take more large rhombi or more small rhombi to cover the yellow area on
page 2? Why?
How many triangles meet at a point when they are used to tile a plane? What about
other shapes?
Did everyone use the same number of shapes in their tessellations? Why might the
number of shapes vary among students?
Among classmates, how many different arrangements were discovered for tessellating
triangles? Squares? Hexagons? Trapezoids? Large Rhombi? Small Rhombi?
For tessellations with 2 or more shapes (pages 4-5)
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Were there different arrangements of shapes or did all students come up with the
same arrangement? Did any arrangements use a repeating pattern?
Which three shapes did students choose for their final tessellation? Are there any
three shapes that were tried but did not tessellate?
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Tessellations
8. Conclude the lesson by asking students to write definition of tessellation or tiling of the
plane. Ask them to record two places where tessellations are used in real life (quilts,
bathroom and kitchen tiles, carpets, art, etc).
Assessment
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Assess students’ problem-solving process and completed tessellations during
independent work.
Assess completed activities. Check that students have covered the yellow area
completely without gaps or overlaps and filled in the blue answer boxes.
Students can be assessed on their contributions to class discussion questions and
their written answers from Step 8.
Lesson Adaptations
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Challenge students to come up with multiple arrangements for each tessellation. For
more space, students can add pages by selecting the Go to Page button.
For kinesthetic learners, provide physical manipulatives at their computer station.
To save time, assign different groups or individuals specific tessellations to complete.
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