This unit examines tessellations, that is, ways of covering the plane (a two-dimensional surface) with copies of the same shape without any gaps or overlaps. Students will investigate what properties shapes must have in order to tessellate. The tessellations investigated involve both non-regular and regular polygons.
Tessellations can be found in a variety of contexts, including in kitchen and bathroom on tiles, linoleum flooring, patterned carpets, parquet wooden floors, and in cultural patterns and artworks. They also demonstrate an application of some of the basic properties of polygons.
Tessellations have other, practical uses. Brick walls are made of the same shaped brick repeatedly laid in rows (a tessellation of rectangles). Bees use a basic hexagonal shape to manufacture their honeycombs (a tessellation of regular hexagons). These tessellations provide a strong structure for their two different purposes.
A key features of tessellations is that the vertices of the figure, or figures, must fit together, meaning that there are no gaps or overlaps in the pattern created, and that the pattern completely covers a given two-dimensional space. This can be achieved in two ways. Either the corners of the basic shape all fit together to make 360° , or the corners of some basic shapes fit together along the side of another to again make 360°. Therefore, a necessary precursor to this unit is a lesson or series of lessons that give the class a sound knowledge of angles in degrees. You might use Measuring Angles, Level 3 for this purpose. The Problem Solving lesson Copycats, Geometry, Level 3 could also be used as part of the Exploring stage of this unit
Fitness, Level 4 follows on from this unit and looks at both regular and non-regular tessellations. In the regular case it shows that regular tessellations can be made only with equilateral triangles, squares and regular hexagons. Semi-regular tessellations involve two or more regular polygons. This unit also investigates the possibility of non-regular tessellations.
Related to the idea of tessellations is that of Escher drawings. There is a unit on that at Level 4, Tessellating Art, though some of the concepts there would be accessible to students at Level 3.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
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. 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, or by the work of New Zealand artist Glen Jones or Australian artist Bruce Bilney. Tessellation might fit well with efforts to beautify the school environment. Mosaic tiles can be created from fired clay, or cobblestones created from concrete. Look for examples of tessellations in students’ environment such as lino, or tile patterns, facades of buildings, or honeycombs in beehives. Look online for examples of tessellation in the natural and human-made world.
Roam and work with the students. If needed, they could work in pairs or small groups. Once they have one brick pattern, see if they can find others. Emphasise that the bricks don’t always have to be ‘horizontal’ and that the pattern doesn’t have to be practical. It’s OK if the pattern won’t produce a very strong wall.
In this session the students will explore tessellations by regular polygons. The key finding is that regular polygons don’t tessellate. If you haven’t got any solid regular polygons handy then they can be made using Copymaster 3, or displayed digitally.
What we want to do is to see which of these shapes tessellate and which don’t. We’ll then construct this table together.
Shape | Tessellate? | Why? |
equilateral triangle | yes | |
square | ||
regular pentagon | ||
regular hexagon | ||
regular heptagon | ||
regular octagon | ||
regular nonagon | ||
regular decagon | ||
regular hendecagon | no | |
regular dodecagon |
Bring the class together to discuss their results. After each group has reported and justified their claims add them to the table. A completed form of the table is given below. The angles talked about in the third column are the interior angles of the polygons. If your students are confident in measuring angles, they could measure the angles of the shapes they investigated and add this information to their charts. Otherwise, briefly model the measurement of the angles in a few shapes, and explain how the total sum of the angles is found (i.e. add them together). If your students are not familiar with measuring angles, and you feel this knowledge will be too much in addition to the learning in this unit, you could use the existence of gaps in the pattern of shapes to justify whether the shape does or does not tessellate
Shape | Tessellate? | Why? |
equilateral triangle | yes | show pattern; 6 x 60° = 360° so they fit. |
square | yes | show pattern; 4 x 90° = 360° so they fit. |
regular pentagon | no | no multiple of 108° makes 360° nor do multiples of 108° plus 180° . |
regular hexagon | yes | show pattern; 3 x 120° = 360° so they fit. |
regular heptagon | no | no multiple of 128.57° makes 360° nor do multiples of 128.57° plus 180° . |
regular octagon | no | no multiple of 135° makes 360° nor do multiples of 135° plus 180° . |
regular nonagon | no | no multiple of 140° makes 360° nor do multiples of 140° plus 180° . |
regular decagon | no | no multiple of 144° makes 360° nor do multiples of 144° plus 180° . |
regular hendecagon | no | no multiple of 147.27° makes 360° nor do multiples of 147.27° plus 180° . |
regular dodecagon | no | no multiple of 150° makes 360° nor do multiples of 150° plus 180° . |
In this session the students try to extend their results about regular polygons to more general polygons. The key findings are that regular polygons don’t tessellate and all quadrilaterals tessellate.
Teaching Notes:
The students will discover that any quadrilateral will tessellate. Because the interior angles of any quadrilateral add up to 360° , we need to put four figures together at a point so that each one of the four (possibly) different angles is used. In this pattern, sides of equal length also have to fit together. The pattern below shows how to do this.
We have shown that we can fit four of these polygons together at a vertex without gaps. How can we be sure that we can continue the pattern indefinitely? You can see that the pattern in the figure is made up of a wiggly strip of quadrilaterals. In this strip, one quadrilateral is placed one way and then it’s placed another. These wiggly strips can put side by side forever. What you see in the drawing is what you would see anywhere in the plane.
Notice that it doesn’t matter whether the quadrilateral is convex (no interior angles bigger than 180° ), as above, or concave, as below.
If all quadrilaterals tessellate is the same true for all triangles? Repeat the last session but this time use triangles. Provide students with a range of triangles to experiment with (e.g. scalene, equilateral, isosceles, right-angle, obtuse, acute). Briefly introduce the names of the triangles you include, and if appropriate, make reference to the types of angles they demonstrate.
Teaching Notes:
It is easier to establish whether or not triangles tessellate. Since the interior angles of a triangle add up to 180° , we need to make sure that each angle is represented at a point. In the diagram below, the angles form a continuous, straight line. This means that we can put the triangles together to make a row. We can fit two such rows together. In fact we can fit as many rows together as we like until the entire plane is covered.
Review the facts: all triangles tessellate; and all quadrilaterals tessellate. What do we know about regular polygons? (They don’t tessellate.) Does that mean that no pentagon tessellates? Ask the students to experiment using the pentagons from Copymaster 5 and by inventing their own pentagonal shapes. What results can you find?
If the students can’t find any pentagonal tessellations, you might remind them how they constructed the triangular and quadrilateral tessellations. Rows were very important. So it might be an idea to try to use rows in some way to try to form pentagonal tilings.
The situation with pentagons is more complicated than with either triangles or quadrilaterals. Some pentagons do tessellate and some do not. From the evidence of the regular pentagon, it is unlikely that all the interior angles of a pentagon could cluster around a similar point in a similar way. That means that there have to be at least two types of points where pentagons come together. So we have to distribute the sizes of the interior angles so that this can happen. This enables us to get tilings like the one below.
Teaching Notes:
There are other hexagons apart from regular hexagons that do tessellate, such as the concave hexagon shown below.
If we can arrange the angles of a hexagon in such a way that no combination of them will add up to 360° (or 180°), then it won’t be possible for that hexagon to tessellate.
The conjecture that only regular polygons that tessellate are the equilateral triangle, the square, and the hexagon is proved in Fitness, Level 4.
Dear family and whānau,
This week in maths we have been looking at tessellations of the plane by different shapes. Your child will be able to tell you what ‘tessellation’ means.
It would be appreciated if you could help your child look around your house (e.g. in artworks and tiling) and local neighbourhood to see if they can find any of the tessellations that we have been talking about. Your child should then make a sketch so that we can talk about them next week.
Here is a challenge: We also know that we can’t use a regular octagon (like a stop sign) to tessellate by itself. But together can you find a non-regular octagon (any 8 sided shape) that will tessellate? A non-regular octagon will have eight sides of unequal length and eight interior angles of unequal size.
Printed from https://nzmaths.co.nz/resource/keeping-shape at 9:13pm on the 13th May 2024