'Discover the mysteries behind beautiful tiling patterns'

The headquarters of the United States Department of Defense is The Pentagon. The shape is based on a regular pentagon and it houses the largest floor space office area in the world.  You might like to consider the advantages of using a pentagonal rather than rectangular prism shapes. Whilst aesthetically interesting, the choice of shape does create some problems.  Can you think of any other buildings made in this shape? Tiling patterns show that other shapes fit together far more efficiently.  Look at the tiling patterns below and start thinking about what factors affect the way that they fit together without leaving any gaps. Then complete the series of activities and discover the geometrical reasons.

Resources

You will need to use dynamic geometry software for the investigation.  Geogebra is simple, free and requires no installation. I recommend certain settings for optimal use in this investigation.  Follow these quick instructions.

This  investigation uses dynamic geometry to help students discover the formula for the sum of interior angles in an n-sided polygon.

Once the formula has been discovered it is time to convince ourselves that this rule is true.  Complete the following  justification.

Using your knowledge about angles in polygons you can create beautiful tiling patterns.  Try this  application.  You may find this nlvm manipulative useful to create the patterns.

Description

The following is an outline of how the activity might run.

• Explore some beautiful tiling patterns and discuss why they fit together and what makes them aesthetically appealing.

• Recall angle facts about triangles and quadrilaterals.  If possible discuss other angle facts, e.g. angles at a point, on a straight line, …

• Open Geogebra and go through the instructions for optimizing the settings for use in this investigation. Setting the angle measure to round to the nearest angle is particularly helpful for students.

• Investigate the angle sums of different polygons.  A calculator may be useful.

• Record the results in a table and take a screenshot of the Geogebra sketches as a visual reminder.

• Students might need some help to generalize their results.

• Use the second worksheet to justify the rule for an n-sided polygon.  There are two methods for doing this.

• Both justifications will give the same result, but some algebraic manipulation will be required to show that they are the same.

• Tessellating regular polygons provides a lovely application of why interior angles of polygons are useful and important.

• The activity is completed with a challenge to find all 9 semi-tessellating polygons.

• A further extension could be to look at 3D tessellations, e.g. the shapes that fit together to make a football.

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