Keeping in Shape

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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.

Achievement Objectives
GM3-6: Describe the transformations (reflection, rotation, translation, or enlargement) that have mapped one object onto another.
Specific Learning Outcomes
  • Demonstrate why a given tessellation will cover the plane.
  • Create regular tessellations.
Description of Mathematics

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.

Opportunities for Adaptation and Differentiation

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:

  • providing physical (or digital) manipulatives and regular polygons so that students can experiment with shapes. Encourage anticipation of results, e.g., “Will regular hexagons tessellate? How do you know?”
  • asking students to justify why they believe patterns occur
  • explicitly teaching angle as a measure of turn through physical action, at first, then diagram drawing, and the use of a protractor to measure the amount of turn
  • directly modelling examples of tessellations and explaining why the combinations of shapes around each vertex will work
  • organising the data about regular polygons in a table, especially the measures of internal angles. Make the table accessible to students so they can make predictions about sets of shapes that will, and will not, tessellate
  • encouraging students to work collaboratively in partnerships, and to share and justify their ideas. Regular sharing of ideas will encourage students to extend their thinking and will allow them to learn from each other. 

The difficulty of tasks can be varied in many ways including:

  • allowing access to calculators and digital tools, so the investigations are more about spatial reasoning than calculation
  • using students’ knowledge of everyday contexts to teach angles as a measure of turn (e.g. around 180s and 360s in skateboarding or surfing)
  • restricting the set of shapes at first, e.g. triangles, squares, pentagons, and hexagons, until generalisations about angles around a vertex have been developed. Then open the investigation to more complex regular polygons
  • displaying students' work as a model for others, especially from students who provide explanations about why particular tessellations work. Motivate students to add a new, undiscovered tessellation to the class display.

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.

Required Resource Materials
  • Copymaster 1 (picture showing brick wall)
  • Copymaster 2 (a sheet of 2 x 1 bricks)
  • Copymaster 3 (a sheet of regular polygon shapes: equilateral triangle, square, pentagon, hexagon, heptagon, dodecagon) 
  • Copymaster 4 a sheet of 4-sided figures: a parallelogram, kite, trapezium, and dart)
  • Copymaster 5 (a sheet of three types of pentagons, two of which tessellate and one which does not
  • A variety of images (e.g. artworks, outdoor structures, nature) showing tessellation
  • Class chart (paper or digital) 
  • Regular polygonal shapes
  • Scissors, paper, rulers, pencils, protractors, Lego
  • Graph paper

Session 1

  1. Talk to the students about buildings and how they are constructed. Give them images of buildings to inspire them and for them to discuss. You could also ask a local builder who is a member of your community to come and talk to your class about construction.
    What material is used to construct buildings? (wood, bricks, mortar, sand, cement, etc.)
    How do builders put a house together?
    How do they build walls?
  2. Get the students to describe the brick pattern on walls. Show them Copymaster 1, or another picture of a brick wall. Talk about the features of the pattern (i.e. there are no gaps, the bricks are in rows, each row is slightly displaced from the row below.)
    Why do you want to have no gaps in a wall?
    (You might note that some walls do have gaps. Why?)
  3. Are there any other ways of using bricks to build a wall?
  4. Get the students to produce their own (2-dimensional) bricks. The bricks should be rectangles that are twice as long as they are wide. This can be done using graph paper and then tracing round one to produce many. Model this process for the class. They could also draw a number of bricks on graph paper to be photocopied, before cutting up the photocopied bricks. Copymaster 2 can be photocopied if you would prefer students not to make their own bricks.
  5. Ask the students to create a wall using the bricks that they have just produced. They might copy the pattern shown in Copymaster 1 or arrange the bricks in any pattern they like. However, if they think up their own patterns there are two rules that must be followed: (i) there are to be no gaps and (ii) the pattern must be capable of being continued indefinitely (to cover a very big wall!). Draw attention back to the brick wall example shown previously, and highlight each of these features.
  6. 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.

    A tessellating pattern constructed from rectangles.

  7. Bring the class together and let the students discuss the patterns that they have made. The students should be able to identify that their patterns have no gaps and that they can go on forever in all directions. Support them with their explanations as necessary. 
  8. Tell the class that they have been making tessellations. A tessellation is a way of using a fixed shape to cover the whole area of a flat surface (a plane). Tessellations have two important properties: (i) they have no gaps (all of the plane is covered) and (ii) they go on forever (no matter where you go in the plane the shapes will still be covering the part of the plane that you can see. Sometimes we call a tessellation a tiling. Record a summary of these key points on a class chart.

Session 2: Regular Polygons

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.

  1. Ask the students to recap the work of the last lesson. They can share their ideas with a buddy, small group or the whole class.
    What did we do in the last lesson?
    What shapes did we use?
    What word did we learn? (tessellation or maybe tiling)
    How many rules does a tessellation have? What are they?
    Display the class chart constructed in the last session, and revise any key points that students didn’t emphasise in discussion.
  2. Today we are going to look at special types of shapes called regular polygons. 
  3. Show the students a variety of relevant artworks and images that feature tessellations (this could be an opportunity to look at a local tapa cloth or kowhaiwhai panel). Ask them to identify any regular polygons. Emphasise that a regular polygon has sides that are equal and interior angles that are equal. 
  4. What we want to do is to see which of these shapes tessellate and which don’t. We’ll then construct this table together.

    equilateral triangleyes 
    regular pentagon  
    regular hexagon  
    regular heptagon  
    regular octagon  
    regular nonagon  
    regular decagon  
    regular hendecagonno 
    regular dodecagon  
  5. Distribute the regular polygons to students and ask them to find out which shapes tessellate and which don’t. Let the students tackle the problem in any way that they like. As you go round their groups check that they understand that there are to be no gaps and that the patterns must continue forever in all directions. Check too, that if they can’t find a tessellation, they can explain why one doesn’t exist. (It isn’t good enough to say that they have tried for 5 minutes and they can’t find one. They need to be able to justify why there have to be gaps or why the pattern can’t go on forever. It should usually be the first reason. This is fortunate, as it is easier to justify.) They could record the findings from their investigations on a large sheet of paper - on which they could draw each tessellation and write a few sentences explaining why it does or does not tessellate. Encourage the students to refer back to the class chart constructed in session 1 for a reminder of the ‘rules’ tessellations must follow.
  6. Groups who finish early could be asked to see if they could find some different tessellations using a combination of the regular polygons, or using the regular polygons they have not yet investigated.
  7. 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

    equilateral triangleyesshow pattern; 6 x 60° = 360° so they fit.
    squareyesshow pattern; 4 x 90° = 360° so they fit.
    regular pentagonnono multiple of 108° makes 360° nor do multiples of 108° plus 180° .
    regular hexagonyesshow pattern; 3 x 120° = 360° so they fit.
    regular heptagonnono multiple of 128.57° makes 360° nor do multiples of 128.57° plus 180° .
    regular octagonnono multiple of 135° makes 360° nor do multiples of 135° plus 180° .
    regular nonagonnono multiple of 140° makes 360° nor do multiples of 140° plus 180° .
    regular decagonnono multiple of 144° makes 360° nor do multiples of 144° plus 180° .
    regular hendecagonnono multiple of 147.27° makes 360° nor do multiples of 147.27° plus 180° .
    regular dodecagonnono multiple of 150° makes 360° nor do multiples of 150° plus 180° .
  8. Ask the class if they have any new information or conjectures (i.e. statements that have not yet been proven as correct or incorrect) about tilings by regular polygons that could be added to the class chart.

Session 3: Quadrilaterals

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.

  1. Get the class to recall what happened in the last session. The key points are (a) what is a tessellation; (b) tessellations were found for only three types of regular polygons; and (c) which types? The key points are (a) what is a tessellation; (b) tessellations were found for only three types of regular polygons; and (c) which types? Display the class chart and highlight any information which students did not identify.
  2. Ask "which shapes do we now know will tessellate?" (The three regular polygons and the rectangle – from the brick patterns.) We know that two kinds of 4-sided figures will tessellate. What other types of quadrilaterals will tessellate?
  3. Provide the students with various quadrilaterals (see Copymaster 4) and let them experiment. A table can again be constructed like the one in Session 1.
  4. Give the students the opportunity to experiment for themselves with other quadrilaterals. (For instance the students who finish quickly and can justify what they have done should go on to inventing quadrilateral tilings of their own.) Ask them to record their findings on the chart they constructed in the previous session. Remind them that there are two things that can be varied – the lengths of the sides and the sizes of the interior angles. Suggest that they vary the interior angles first. If your students are not familiar with measuring angles, ask them to look at the lengths of the sides, and whether any gaps are formed when they try to tessellate the new shapes.
  5. Let the students report back on what they found. What new information did they discover? What conjecture do they have about tessellations by quadrilaterals? Can they justify that conjecture?

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.
Tessellating quadrilaterals.

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.Tessellating convcave quadrilaterals.

Session 4: Triangles.

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. Tessellating triangles form a row.

Session 5: Pentagons

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.
Tessellating pentagons.

Session 6: Reflecting

  1. Discuss with the class the results of the last few days. Refer back to the charts constructed and record some summary statements.
    What have you discovered?
    What is a tessellation?
    What polygons do you know that tessellate?
    What polygons definitely do not tessellate?
  2. Now we know that all triangles tessellate, all quadrilaterals tessellate and that regular hexagons tessellate.
    What questions do you think that we ought to ask now? (Do you think that all hexagons tessellate? What do you think? Why?)
  3. Experiment for yourselves. Make your own hexagonal shapes and come up with a conjecture.
  4. After they have tried this for a while, get them to report back on their conclusions.
  5. Revisit their conjecture about regular tessellations. How could you prove this?

Teaching Notes:
There are other hexagons apart from regular hexagons that do tessellate, such as the concave hexagon shown below.

A concave hexagon.

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.

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