TESSELLATING
THE GEOBOARD
Getting Ready
What You’ll Need
GEOMETRY
• Tessellation
• Congruence
• Spatial visualization
Overview
Children investigate ways to fit shapes together to cover their Geoboards
without gaps or overlaps. In this activity, children have the opportunity to:
Geoboards, 1 per child
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begin to understand the concept of tessellation
Geodot paper, page 90
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discover how certain shapes tessellate
Overhead Geoboard and/or geodot
paper transparency (optional)
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realize that some shapes tessellate in more than one way
Rubber bands
The Activity
Introducing
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Prepare a Geoboard that shows a tessellation of one-by-one squares.
Keep it face down.
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Have children make the smallest possible square they can on their
Geoboards.
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Ask children whether, if they were to cover the entire Geoboard with
more squares this size, there would be any gaps.
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Give children time to work on this problem and then share their
responses.
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Show a prepared Geoboard to confirm that there would be no gaps.
Explain that when you use the same shapes over and over again
to cover a surface with no gaps or overlaps, you have tessellated
that surface.
Point out that the unavoidable rubberband loop is considered a single line
segment; therefore, these squares do
not overlap.
© ETA/Cuisenaire®
On Their Own
Can you find different ways to tessellate your Geoboard?
• Completely cover your Geoboard with 2-by-2 squares.
• Make sure that the squares you make don’t overlap or have gaps between them.
• Record your tessellation on geodot paper.
• See if the 2-by-2 squares can tessellate in other ways. If so, record each
tessellation.
• Repeat the process for each of the following shapes:
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a 1-by-2 rectangle
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a 1-by-4 rectangle
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a triangle with an area of 1 square unit
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a triangle with an area of 2 square units
• With your group, discuss how the different shapes tessellate.
The Bigger Picture
Thinking and Sharing
Create five column headings across the chalkboard, one for each kind of shape used. Have
volunteers post their work in the appropriate columns, checking that each posting is not just
a flip or rotation of an already-posted tessellation.
Use prompts such as these to promote class discussion:
◆
What do you notice when you look at the posted shapes?
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Which shapes tessellate in only one way?
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Why can some shapes tessellate in more than one way while others can’t?
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Do you think there might be Geoboard shapes that won’t tessellate? Explain.
Extending the Activity
1. Have children make another shape and figure out different ways that it will tessellate
their Geoboards.
2. Ask children to try to use a combination of two shapes to tessellate their Geoboards.
© ETA/Cuisenaire®
Teacher Talk
Where’s the Mathematics?
This activity provides an opportunity for children to investigate tessellations
with rectangles and triangles. As they experiment, children discover that the
two-by-two square and the one-by-four rectangle tessellate the Geoboard in
only one way.
2-by-2 squares
1-by-4 rectangles
In contrast, the one-by-two rectangle tessellates the Geoboard in many ways.
1-by-2 rectangles
Tessellating with the triangles is more challenging than tessellating with the
rectangles. Children must first make a triangle that has the given area. As it
happens, two different triangles with an area of 1 square unit will tessellate
the Geoboard. One of these is an isosceles triangle.
Isosceles triangles each with an area of 1 square unit
The other is a scalene triangle.
Scalene triangles each with an area of 1 square unit
© ETA/Cuisenaire®
There are also two different 2-square-unit triangles, again, one isosceles and
one scalene, that will tessellate the Geoboard. Each tessellates the Geoboard
in more than one way.
Isosceles triangles, each with an area of 2 square units
Scalene triangles, each with an area of 2 square units
In looking at the posted shapes, children may observe that the triangles
that they used to tessellate the Geoboard can be put together to form the
squares and rectangles they used to tessellate the Geoboard. This may help
explain why these triangles tessellate. In particular, the isosceles triangles
can be joined to form squares and the scalene triangles can be joined to
form rectangles.
Children do not have to use the terms
isosceles and scalene; however, by
mentioning them naturally as they discuss the triangles, you give children
the language that can help them
communicate more effectively.
As children experiment to find ways to tessellate their Geoboards, they gradually develop an understanding of the mathematical relationships that must
exist in shapes that tessellate. So that gaps or overlaps do not occur, the sum
of the angles of the shapes at the point where they meet must be 360°.
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90 90
90 90
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© ETA/Cuisenaire®