Project AMP
Dr. Antonio Quesada – Director, Project AMP
Tessellations
Lesson Summary:
Students will create polygon tessellations by rotation using geometry software.
Students will work with both regular and irregular polygons.
Key Words:
Tessellations, regular polygons, rotation
Existing knowledge:
Students should have previous knowledge of regular polygons, interior angles in a
polygon, and the midpoint of a segment.
Learning objectives:
1. Create regular polygon tessellations by rotation.
2. Determine which regular polygons tessellate.
3. Create polygon tessellations by rotation.
4. Determine what conditions cause a polygon to tessellate.
Materials:
Computer lab or set of calculators equipped with Cabri Geometry II and a lab worksheet.
Suggested Procedure:
• Group students in pairs.
• Discuss the definition of a tessellation with students.
• Have students complete the lab to discover the properties of tessellations.
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Dr. Antonio Quesada – Director, Project AMP
Lab: Constructing Tessellations
Team Members’ Names ________________________________________________
File Name _______________________________________________
Investigate Using Cabri Geometry II
Tessellations are patterns that cover a plane with repeating figures so there is no gaps
or overlapping. A common example of a tessellation is tile floors. Tessellations can
be made up of a tiling of one or more figures. However, in this investigation only
monohedral tiling, or a tiling of only one figure, will be explored.
Goal 1: Construct regular polygon tessellations.
1.
Create a regular triangle.
What is each interior angle measure? __________
[Use regular polygon tool.]
2. Rotate the triangle around one vertex by the interior angle measure. Refer to
appendix to recall this procedure. Continue this process until you are able to
determine whether this polygon tessellates the plane. Remember: there should be
no gaps or overlapping.
[Use Numerical Edit, and Rotate tools.]
3. Repeat the directions in #2 for a regular quadrilateral, a regular pentagon, and a
regular hexagon.
4. Which of the regular polygons above tessellate?
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5. Explain using angle measures the conditions required for a regular polygon to
tessellate.
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__________________________________________________________________
Goal 2: Construct polygon tessellations.
1. Draw a scalene triangle.
a. Find the midpoint of each side.
[Use Midpoint tool.]
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Dr. Antonio Quesada – Director, Project AMP
b. Rotate the triangle about any midpoint 180o . Refer to appendix to recall
this procedure. Continue this process until the screen is filled or the
polygons overlap.
[Use Midpoint, Numerical Edit, and Rotate tools.]
c.
Does the triangle tessellate? Explain your conclusions using your
knowledge of the sum of the interior angles of a triangle.
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2. Draw a convex or concave quadrilateral.
a. Follow the same process described in #1.
b. Does the quadrilateral tessellate? Explain your conclusions using by using
your knowledge of the sum of the interior angles of a quadrilateral.
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3. It is possible for some pentagons to tessellate. Repeat the previous process for a
pentagon. If the pentagon you create does not tessellate, change the shape using
the hand tool until the pentagon does tessellate. When creating the pentagon,
label the vertices to distinguish the original pentagon.
Why do some pentagons tessellate when others do not?
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Appendix:
To create a regular triangle, select the Regular Polygon tool. Click on the screen to
create the center point. Rotate your mouse clockwise until a regular triangle appears,
then click. Select the Numerical Edit tool from the display toolbox. Type in the interior
angle measure of the regular polygon. Press CRTL + U and select the degrees unit.
Select the Rotate tool from the transform toolbox. Click on the polygon, click on the
angle measure, then click on one of the vertices of the polygon. Repeat this process on
the newly constructed polygon.
To rotate the polygon 180o about a midpoint, select the Numerical Edit tool from the
display toolbox. Type in the value 180. Press CTRL + U to see a list of units. Select
degrees. Select the Rotate tool from the transform toolbox. Click on the polygon, click
on the numerical value, then click on the midpoint of any side. Repeat the process by
finding the midpoint of one of the new sides created and rotating about that point.
Extension
Your grandmother has asked you to buy marble tile imported from Italy for her kitchen
floor. The dimensions of her rectangular kitchen are 12’ × 15’. Your grandmother wants
to compare the price of using square tiles versus hexagonal tiles. The length of one side
of a square tile is 6’’ with a cost of $3.50 a piece and the length of one side of a
Project AMP
Dr. Antonio Quesada – Director, Project AMP
hexagonal tile is 6’’ with a cost of $8.00 a piece. Which type of tile is a better buy?
Explain.
Using the scale factor of 1 cm = 1 foot, use Cabri to create a scale drawing of the kitchen
to determine the necessary number of tiles in each case.