The Tessellation Tutorial

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The Tessellation Tutorial
Have you ever looked at your shower wall?
Or maybe even your kitchen floor?
Did you ever notice any special design to it?
Well the pattern you see is known as a
tessellation. This lesson explores what a
tessellation is and the many features that make
up the design of the tessellation. Get out a
pencil and paper to follow along with the
activities. Follow the program and see if you can
create your own tessellation by the end of the
lesson!
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Table of Contents
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Tessellations

• What is a tessellation?
• A tessellation is created when a shape is
  repeated over and over again covering a plane
  without any gaps or overlaps.
• Another word for tessellation is a tiling.
• How is a tessellation made?
• Tessellations follow a set of
• A tessellation can be made by two patterns
• Regular Tiling Semi-Regular Tiling

Rules.
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The Rules

• RULE 1   The tessellation must tile a floor
  (that goes on forever) with no overlapping or
  gaps.
• RULE 2  The tiles must be regular polygons.
• RULE 3   Each vertex must look the same.
• A vertex is a point at which all the corners
  meet.
• The interior angles of the polygons must add up
  to 360o at any vertex.

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Interior Angles of Polygons

• We know that the sum of the measures of the
  interior angles of any triangle is 180o.
• In an equilateral triangle, all the sides are the
  same length, which means that all the angles are
  equal.
• The formula to find the measure of each angle is
• 180 60
• 3
• Therefore, each angle is 60o.
• Try some examples next. Check the answer key to
  see if you are right.

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Finding interior angle measures

• How do you find the interior angle measures of
  other polygons?
• Step 1 Take any polygon and divide it into
  triangles.
• Step 2 Count the number of triangles that are
  formed within each polygon.
• Step 3 Multiply the number of triangles formed
  by 180o. This gives the total number of degrees
  in the polygon.
• Step 4 Divide the product by the number of sides
  in the polygon. This gives the interior angle
  measure of each angle.
• Take some time and try a few. When you finish,
  check the

answer key.
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Activity
Step 1 Step 2
Step 3 Step 4
A square can form 2 triangles.
A pentagon can form 3 triangles.
A hexagon can form 4 triangles
Answers
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Step 1 Step 2
Step 3 Step 4
A square can form 2 triangles.
2 x 180 360 360 90o
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A pentagon can form 3 triangles.
3 x 180 540 540 108o
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A hexagon can form 4 triangles.
4 x 180 720 720 120o
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The Chart Activity


Fill in the chart with your answers and then
check the . This will help you
understand the next sections.
answer key
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Regular Tiling

• A regular tessellation means a tessellation made
  up of only congruent regular polygons.
• Remember Regular means that the sides of the
  polygon are all the same length. Congruent means
  that the polygons that you put together are all
  the same size and shape.
• Look at the next slide and make some predictions
  as to which polygons will form regular
  tessellations.
• Use your chart to make some predictions as to
  which polygons form regular tessellations.

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Polygonal Shapes
Triangle Square Pentagon
Hexagon Heptagon
Make some predictions then try tessellating some
of these shapes at (website). Did it work? Which
shapes tessellated by the rules? Click on a shape
to learn more
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The Equilateral Triangle
Yes! Triangles form a regular tiling.

• We can tessellate triangles by following the
  rules.
• You found that the interior angle of an
  equilateral triangle is 60o. We see that there
  are six triangles that meet at each vertex.
• 60 60 60 60 60 60 360o.

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The Square
Yes! Squares form a regular tiling.

• We can tessellate squares by following the
  rules.
• You found that the interior angle of a square is
  90o. We see that there are four squares that meet
  at each vertex.
• 90 90 90 90 360o.

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The Pentagon
No! Pentagons will not work.

• We cant tessellate pentagons by following the
  rules.
• Which rule(s) doesnt the tessellation follow?
• You found that the interior angle of a pentagon
  is 108o. We see that there are three pentagons
  that meet at each vertex.
• 108 108 108 324o.

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The Hexagon
Yes! Hexagons form a regular tiling.

• We can tessellate hexagons by following the
  rules.
• You found that the interior angle of a hexagon
  is 120o. We see that there are three hexagons
  that meet at each vertex.
• 120 120 120 360o.

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The Heptagon
No! Heptagons will not work.

• We cant tessellate heptagons by following the
  rules.
• Which rule(s) doesnt the tessellation follow?
• You found that the interior angle of a heptagon
  is 128o. We see that there are three heptagons
  that meet at each vertex.
• 128 128 128 384o.

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Assessment
In a tessellation the polygons used will fit
together with their angles arranged around a
point with no gaps or overlaps. When using just
one polygon (for example, only equilateral
triangles), the interior measure of each angle
will need to be a factor of _____ degrees
(meaning that ____ degrees can be divided evenly
by that angle measure). The only regular polygons
that qualify are the __________________,
___________________, and ___________________.
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Semi-regular Tessellations

• A semi-regular tiling is a tessellation that is
  made up of two or more regular polygons at a
  vertex.
• The rules are still the same for semi-regular
  tessellations.
• The pattern must be the same at each vertex.
• The fewest number of polygons that can be at a
  vertex is 3 and the most number of polygons that
  can be at a vertex is 6.
• Lets try some finding some combinations next.
• There are 17 ways to combine polygons at a
  vertex following the proof
• 3 polygons at a vertex 6

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3 polygons at vertex
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• The combination of polygons that work is 1 square
  and 2 octagons.
• 1 square 90o
• 2 octagons 2 x 135o 270o
• 270 90 360o

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Other combinations
Here is one with 4 polygons at a vertex 2
triangles? 2 x 60 120 2 hexagons? 2 x 120
240 120 240 360
Here is one with 5 polygons at a vertex 3
triangles? 3 x 60 180 2 squares? 2 x 90 180
180 180 360
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Assessment

• Look at the tiling at the right. Answer the
  following question
• Is this a tessellation?
• If yes, which type is it and why?
• If not, what makes it not a tessellation?

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The 4 symmetries
Symmetry is an important concept when working
with tessellations. Click on an icon to learn
more.
Translation Reflection
Glide Reflection Rotation
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Translation
In this case, triangle ABC has been translated
3 units to the right and 4 units up.

• To translate an object means to slide it up,
  down, left, or right.
• The shape itself does not change in any way.
• Every translation has a direction and a distance.
  
• The arrow that marks the direction is called a
  vector.

Click to explore
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Reflection
Notice how point B (the original shape) and
point B (the reflected shape) are the same
distance from the mirror line. They are both 5
units from the mirror line.

• To reflect an object means to produce its mirror
  image.
• Every reflection has a mirror line.
• The original object and its image should be the
  same distance apart.

Click to explore
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Glide Reflection

• Glide reflections are the only type of symmetry
  that involve more than one step.
• A glide reflection combines a reflection with a
  translation along the direction of the mirror
  line.

Walking is the perfect example of a glide
reflection.
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Rotation
This isnt science class but rotation is seen
when the Earth goes around the sun.

• A rotation is a transformation that turns a
  figure about a fixed point called the center of
  rotation. 
• An object and its rotation are the same shape and
  size, but the figures may be turned in different
  directions.

Click to explore
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Assessment

• Which of the following shapes are translations of
  triangle A?

A B C D E F G H I J K L M N O P Q R S T U V W X Y
Z

• Look at the alphabet to the right. Which letters
  have reflection symmetry? Which ones have
  rotational symmetry?

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Make your own Tessellation

• Now that you have learned all about
  tessellations, try doing some on your own.
• You can create many cool patterns.
• Click on the button to being.

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Teachers

• The purpose of this lesson is to enhance
  students learning and knowledge of
  tessellations.
• There are various concepts that will be regarded
  as review, but important in understanding
  tessellations.
• The Tessellation Tutorial program is designed for
  students in grades 5 and 6.
• The lesson was designed as an independent study
  for the students.

• I have included a lesson plan that should be
  followed before students dive into this
  interactive program.
• Please see the reference page to find more
  helpful links and websites that compliment this
  lesson.

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References
Cool Math
Tessellations
BBC lessons
Rotation symmetry
Symmetry lesson
Math Forum- symmetries
Math Forum- tessellations
Manipulatives