This unit examines regular tessellations, that is, tessellations that can be made using only one type of regular polygon, and semi-regular tessellations, where more than one type of regular polygon is involved. Students are required to investigate what properties tessellating shapes must have in order to cover the plane with no gaps or overlaps.
Tessellations are frequently found in kitchen and bathroom tiles and lino. You can see them in the pattern on carpets and decorative patterns on containers and packaging. They play a significant role in tapa cloth design and creation, and in Islamic art that features designs commonly built around star polygons. Tessellations are a neat and symmetric form of decoration. They also provide a nice application of some of the basic properties of polygons.
To be able to fully understand the concept of tessellations using regular polygons, you need to recognise their symmetry, and be able to calculate the size of their interior angles. This information is accessible to Level 4 students. In this unit, students are led through the steps needed to establish that there are only three regular polygons that tile the plane. This unit follows on from Keeping in Shape from Level 3, where regular tessellations are first discussed.
Moving on from here, the children can consider semi-regular tilings. All that they need to know here is how to sum the interior angles of various regular polygons to 360°. The rest is up to their imagination.
This unit is designed for students to learn and practise outcomes at Level 4 of mathematics in the New Zealand Curriculum. The geometric focus opens up opportunities for visual reasoning that might prove engaging for students who find numeric reasoning challenging. Here are some approaches to enabling participation.
The difficulty of tasks can be varied in many ways including:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Investigate the use of tessellation in cultural designs such as the mosaic art and architecture of the Moors, Greeks, and Persians in Europe and China and Japan in Asia. For example, tessellations are prominent in Islamic art traditions, and in tapa cloth designs from Pacific nations. Students might be fascinated by the work of Dutch artist Escher, who built his work on distorting regular polygons to create ‘life-like’ tessellation patterns. Tessellation might fit well with efforts to beautify the school environment, such as creating a class mural. Mosaic tiles can be created from fired clay, or cobblestones created from concrete.
Show the students a large cut out equilateral triangle. Mark the vertices (corners) of the triangle with different colours. Say, "I am going to tear off the corners of this triangle and place them around this point (draw point on board). What do you think will happen?" Students may have encountered this before but let them guess what they think will happen. Tear off the corners and place them about the point to confirm that a half turn (or 180°) is created.
If the sum of the interior angles of any triangle is the same - 180°, is it likely that the sum of the interior angles of any quadrilateral is the same? Ask them to predict what will happen if the corners of any quadrilateral are joined about a point. Have them cut out several quadrilaterals of different shapes to check their predictions. Remind them to mark the corners before tearing them off.
Once it has been found that the sum of the interior angles of any quadrilateral is 360° (they form a full turn about a point), then students can investigate the sum of interior angles of other polygons by measuring with protractors or tearing corners. You may need to model how to find the size of an angle using a protractor. The results can be captured in the table below. Encourage the students to look for a pattern to predict what the next angular sum will be.
Number of Sides | Sum of Angles ° | Interior Angle of Regular Polygon |
3 | 180 | |
4 | 360 | |
5 | 540 | |
6 | 720 |
Note that for each side that is added, the sum of the interior angles increases by 180°. This can be explained by the fact that the addition of a side creates another triangle within the shape, and that each triangle has an angular sum of 180°.
Quadrilateral (2 triangles) | Pentagon (3 triangles) | Hexagon (4 triangles) |
What size are the interior angles of the regular figures we have just been talking about? How can we find out?
For the regular hexagon with six vertices and an angular sum of 720°, we need to divide the sum by the number of angles to find each angle size. So, since 720 ÷ 6 = 120, each interior angle in a regular hexagon is 120°.
Use a set of pattern blocks to show how equilateral triangles tessellate. Tessellate means that they cover the plane infinitely with no gaps or overlaps. Send the students away in groups with their own set of pattern blocks, or access to an online version (search for Pattern Block Virtual Manipulative), to explore what other tessellations can be discovered using shapes from the set. You may want your students to record the tessellations using isometric dot paper.
After a period of exploration, bring the class back together to share the tessellations. Note that there are three regular tessellations that can be found, that is tessellations involving use of the same regular polygon. These regular tessellations are 3.3.3.3.3.3 (six triangles about each point or vertex), 4.4.4.4 (four squares about each vertex), and 6.6.6 (three hexagons about each vertex).
Focusing on the regular tessellations ask why it is that these patterns work without gaps or overlaps. You may need to remind the students of the angle measures they found in Getting Started. There are two key properties of the shapes involved in regular tessellations:
Dear family and whānau,
This week we have been experimenting with tessellation of the plane with polygons. In particular we have found that regular tessellations can only be made with equilateral triangles, squares and hexagons.
Regular polygons have all sides equal and all angles equal.
Printed from https://nzmaths.co.nz/resource/fitness at 1:11am on the 21st April 2024