CHAPTER
12
Tessellations
GET READY
642
Math Link
644
12.1 Warm Up
645
12.1 Exploring Tessellations With Regular
and Irregular Polygons
646
12.2 Warm Up
652
12.2 Constructing Tessellations Using
Translations and Reflections
653
12.3 Warm Up
658
12.3 Constructing Tessellations
Using Rotations
659
12.4 Warm Up
663
12.4 Creating Escher-Style Tessellations
664
Chapter Review
670
Practice Test
674
Wrap It Up!
676
Key Word Builder
677
Math Games
678
Challenge in Real Life
679
Chapters 9-12 Review
680
Task
688
Answers
690
Name: _____________________________________________________
Date: ______________
Congruent Figures
Congruent figures have the same shape and size.
corresponding sides and angles
equal sides and angles of congruent figures
means is congruent to
ABC
A means angle A.
AB means line segment AB.
DEF
A
D
AB DE
B
E
AC DF
C
F
BC
A
E
+
º
D
B
EF
º
º
+
º
C
F
1. Are the figures in each pair congruent? Circle the correct answer.
P
a)
M
R
Q
L
N
Tick marks
mean the sides
are equal.
b)
congruent or not congruent
C
D
B
A
J
G
I
H
congruent or not congruent
Characteristics of Regular Polygons
regular polygon
a polygon with all equal sides and all equal angles
example: an equilateral triangle
irregular polygon
a polygon that does not have all sides and all angles equal
example: an isosceles triangle
equilateral
triangle
regular
polygon
isosceles
triangle
irregular
polygon
2. Decide if each figure is a regular or irregular polygon.
Circle the correct answer.
a)
b)
STOP
regular polygon or irregular polygon
642
MHR Chapter 12: Tessellations
regular polygon or irregular polygon
Name: _____________________________________________________
Date: ______________
Transformations and Transformation Images
transformation
moves a geometric figure to a different position
examples: translations, reflections, rotations
translations—also called slides
ABC has been translated 4 units
vertically ( ).
The translation image is A B C .
reflections—also called flips or mirror images
Rectangle PQRS has been reflected in the line
of reflection, m.
Rectangle P Q R S is the reflection image.
y
4 P’
y
A
S’
R’
2 line of reflection
m
2
B
C
–2A’ 0 x
–4
0 S
–2
B’
Q’
2 R4 x
–2 P
C’
Q
rotations—also called turns
DEF has been rotated
180º counterclockwise
around the origin.
D E F is the rotation image.
y
centre of
rotation
2
THE is rotated around the centre of rotation, z.
The coordinates of T H E are
(2, –2),
, and
E9
2 x
y
1
.
THE has been rotated
F
–2
E
3.
F9
0
D
D9
2
180°.
E’
–4 –3 –2 –1 0 1 2 3 4 x
–1
T
H –2
z
H’
T’
–3
–4
E
(clockwise or counterclockwise)
(cw)
(ccw)
4. a) On the coordinate grid, translate MON 3 units up and
4 units left.
b) Use the x-axis as the line of reflection to reflect MON.
Use prime notation.
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
–4
2
1
3
M
4 x
O
N
Get Ready MHR 643
Name: _____________________________________________________
Date: ______________
Mosaics are pictures or designs made of different coloured shapes.
Mosaics can be used to decorate shelves, tabletops, mirrors, floors,
walls, and other objects.
You can use regularly and irregularly shaped tiles that are congruent
to make mosaics.
A
C
X
B
Z
Y
a) Measure the sides of each triangle in millimetres.
AC =
mm
AB =
CB =
ZX =
mm
XY =
ZY =
b) Measure the angles of each triangle.
A=
°
X=
B=
C=
Y=
Z=
c) Is ABC congruent to XYZ? Circle YES or NO.
Give 1 reason for your answer.
_________________________________________________________________________
d) Are ABC and XYZ regular or irregular? Circle REGULAR or IRREGULAR.
Give 1 reason for your answer.
_________________________________________________________________________
e)
644
Copy ABC or XYZ onto a piece of cardboard or construction paper.
Cut out the triangle to use as a pattern.
Create a design on a blank sheet of paper.
Trace the triangle template a few times to make a pattern.
Make sure there are no spaces between the triangles.
Colour your design so that your pattern stands out.
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
12.1 Warm Up
1. Fill in the blanks with the word(s) from the box that best describes each diagram.
equilateral triangle
isosceles triangle
pentagon
octagon
square
hexagon
a)
b)
c)
d)
e)
f)
penta means 5
hexa means 6
octa means 8
2. a) Measure the sides and interior angles of the shape.
A
B
An interior angle is
inside the shape.
E
C
D
Angles
A=
Sides
°
AB =
cm
B=
BC =
cm
C=
CD =
D=
DE =
E=
AE =
b) What do you notice about the angles and the sides in this diagram?
_________________________________________________________________________
c) Circle the words that best describe this figure.
regular hexagon
irregular hexagon
regular pentagon
irregular pentagon
12.1 Warm Up MHR 645
Name: _____________________________________________________
Date: ______________
12.1 Exploring Tessellations With Regular and Irregular Polygons
tiling pattern
• a pattern that covers an area or plane with no overlapping or spaces
• also called a tessellation
tiling the plane
• congruent shapes that cover an area with no spaces
• also called tessellating the plane
A plane is a 2-D
flat surface.
Working Example: Identify Shapes That Tessellate the Plane
A full turn = 360°.
Do these polygons tessellate the plane? Explain why or why not.
a)
b)
90º
90º
96º
116º
90º
90º
106º
116º
106º
Solution
Solution
Arrange the squares along a side with
the same length.
Rotate the squares around the centre.
Arrange the pentagons along a side
with the same length.
96º
96º 96º
96º
90º
90º 90º
90º
The irregular pentagons overlap.
They do not overlap or leave spaces.
The shape
The shape
be
(can or cannot)
used to tessellate the plane.
90° + 90° + 90° + 90° =
This more than a full turn.
So, the shape
(can or cannot)
be used to tessellate the plane.
So, the shape can be used to
the plane.
646
96° + 96° + 96° + 96° =
°.
This is equal to a full turn.
(can or cannot)
used to tessellate the plane.
Check:
Each of the interior angles where
the vertices of the polygon meet is
Check:
Each of the interior angles where the
vertices of the polygons meet is 90°.
MHR Chapter 12: Tessellations
be
°.
°.
Name: _____________________________________________________
Date: ______________
Which of the shapes can be used to tessellate the plane?
Give 1 reason for your answer.
a)
120º
b)
60º
120º 120º
120º
60º
120º
120º
120º 120º
Trace the shape and cut it out.
Arrange 4 shapes along a side with
the same length.
Draw the diagram.
Trace the shape and cut it out.
Arrange 3 shapes along a side with
the same length.
Draw the diagram.
120° 60°
60° 120°
Add the interior angles.
+
+
+
Add the interior angles.
=
Is this equal to a full turn?
Circle YES or NO.
Is this equal to a full turn?
Circle YES or NO.
The shape
The shape
(can or cannot)
be used to tessellate the plane.
c)
50º
70º
60º
(can or cannot)
be used to tessellate the plane.
Trace the shape and cut it out.
Arrange 2 shapes along a side with the same length to
make a parallelogram.
Draw the diagram. Label the degrees of each angle.
How many congruent triangles make a parallelogram?
Any triangle
the plane
(will tessellate or will not tessellate)
because congruent triangles always make a
.
(parallelogram or trapezoid)
12.1 Exploring Tessellations With Regular and Irregular Polygons MHR 647
Name: _____________________________________________________
Date: ______________
1. a) Draw a regular polygon that tessellates the plane.
The congruent shapes
must not leave spaces
or overlap.
b) Measure the degrees in each angle.
Write the degrees inside each interior angle of the polygon.
c) Explain why your shape tessellates the plane.
_________________________________________________________________________
2. Does this regular polygon tessellate the plane?
Measure each angle.
Each angle is
°.
Trace this polygon and use it to draw a tessellation.
Add the interior angles.
+
+
=
Is this equal to a full turn? Circle YES or NO.
The polygon
tessellate the plane.
(can or cannot)
648
MHR Chapter 12: Tessellations
A full turn is 360°.
Name: _____________________________________________________
Date: ______________
3. Tessellate the plane with each shape.
Draw and colour the result on the grid.
a)
b)
4. Describe 2 tessellation patterns that you see at home or school.
Name the shapes that make up the tessellations.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5. Jared is painting a mosaic on a wall in his bedroom.
It is made up of tessellating equilateral triangles.
Use the dot grid to draw a tessellation pattern for him.
12.1 Exploring Tessellations With Regular and Irregular Polygons MHR 649
Name: _____________________________________________________
6. Patios are often made from rectangular bricks.
This is a herringbone pattern.
Congruent means
exactly the same.
On the grid, create a different patio design.
Use congruent rectangles.
7. A pentomino is a shape made up of 5 squares.
Choose 1 of the pentominoes.
Make a tessellation on the grid paper.
Use different colours to create an interesting design.
8. Sarah is designing a pattern for the hood of her new parka.
In her design, she wants to use
• a regular polygon
• 3 different colours
Make a design that Sarah might use.
Colour your design.
650
MHR Chapter 12: Tessellations
Date: ______________
Name: _____________________________________________________
Date: ______________
This tiling pattern is from Alhambra, a palace in Granada, Spain.
a) There are 4 different tile shapes in this pattern.
• Circle 1 of each shape in the pattern with a coloured pencil.
• Write the numbers 1 to 4 in each shape.
• Fill in the chart.
Shape
1
Name of Shape
Regular Polygon? Yes/No
2
3
4
b) Trace 6 of each shape on construction paper.
c)
Cut out all 24 shapes.
Use each of the 4 shapes to create a mosaic.
Glue them on another sheet of paper.
Compare your design with your classmates’ designs.
12.1 Math Link MHR 651
Name: _____________________________________________________
Date: ______________
12.2 Warm Up
1. Name each transformation. Use the definitions in the box to help you.
y
a)
6
B
D
A
A’
• A translation is a slide.
4
B’
C
–2
• A rotation is a turn.
D’
2
• A reflection is a mirror image.
C’
0
2
4
x
n
y
b)
c)
6
y
0
4
–2
2
–4
–6
–4
–2
0
2
4
6 x
–2
2. Write the names of the polygons used in each tessellation.
a)
652
MHR Chapter 12: Tessellations
b)
8 x
6
4
2
E
R
N
R’
E’
Name: _____________________________________________________
Date: ______________
12.2 Constructing Tessellations Using Translations and
Reflections
orientation
• the different position of an object after it has been translated, rotated,
or reflected
Working Example: Identify the Transformation
a) What polygons are used to make this tessellation?
the shape that is repeated
in a tessellation
Solution
The tessellation tile is made from the following shapes:
• 2 equilateral triangles
• 1
• 2
b) What transformations are used to make this tessellation?
Solution
A transformation
moves a figure to a different
position or orientation.
This tessellation is made using
.
(translations, rotations, or reflections)
The tessellating tile is translated vertically ( ) and
(
).
c) Does the area of the tessellating tile change during the tessellation?
Solution
The area of the tessellating tile does not change.
The tile remains exactly the same size and shape.
12.2 Constructing Tessellations Using Translations and Reflections MHR 653
Name: _____________________________________________________
Date: ______________
What transformation was used to create this tessellation?
Explain your reasoning by filling in the blanks.
The shapes in the tessellation are
and
.
This tessellation is made using
.
(translations, rotations, or reflections)
1. Jesse and Brent are trying to figure out how this tessellation was made.
Jesse says
Brent says
The tessellation is
made by reflecting
the 6-sided polygon.
The tessellation is made
by translating the 6-sided
polygon horizontally and
reflecting it vertically.
Whose answer is correct? Circle JESSE or BRENT or BOTH.
Give 1 reason for your answer.
_____________________________________________________________________________
_____________________________________________________________________________
654
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
2. Complete the chart.
Tessellation
Names of Polygons in
Tessellation
Type of Transformation
Used
a)
b)
c)
3. Simon is designing a wallpaper pattern that tessellates.
He chooses the letter “T” for his pattern.
Make a tessellation using the 3 letters.
12.2 Constructing Tessellations Using Translations and Reflections MHR 655
Name: _____________________________________________________
Date: ______________
4. The diagram shows a driveway made from irregular 12-sided bricks.
a) Explain why the 12-sided brick tessellates the plane.
Find a point where the vertices meet.
The sum of the interior angles at this point equals
°.
b) On the grid paper, draw a tessellation using a 6-sided brick.
c) Explain why your 6-sided brick tessellates the plane.
_________________________________________________________________________
5. a) Design a kitchen tile.
Use 2 different polygons and translations to make a tessellation.
b) Name the polygons in your tessellation. __________________________________________
c) Name the translations in your tessellation. ________________________________________
656
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
Many quilt designs are made using tessellating shapes.
a) What shapes do you see in the design?
_______________________________
_______________________________
b) The quilt uses fabric cut into triangles.
The triangles are sewn together to form a
.
(name the shape)
c) The squares are translated
( ) and
( ).
d) Design your own quilt square using 1 regular tessellating polygon.
Make an interesting design using patterns and colours.
12.2 Math Link MHR 657
Name: _____________________________________________________
Date: ______________
12.3 Warm Up
1. Draw the regular polygons on the grid.
a) hexagon (6 sides)
c) square
e) parallelogram
b) octagon (8 sides)
d) isosceles triangle (2 equal sides)
f ) equilateral triangle (all equal sides)
2. Circle the diagram(s) that show a rotation.
y
4
y
y
2
2
2
–4
0
2
0
–2
4 x
x
–2
2 x
0
–2
–2
–2
3. Name the polygon(s) in each tessellation.
a)
b)
4. Complete each sentence. Use the words from the box to help you.
a) Another word for slide is
.
b) A turn about a fixed point is called a
c) A
658
MHR Chapter 12: Tessellations
is a mirror image.
.
translation
reflection
rotation
Name: _____________________________________________________
Date: ______________
12.3 Constructing Tessellations Using Rotations
Working Example: Identify the Transformation
a) What polygons are used to make this tessellation?
Solution
The tessellation is made up of regular
.
b) What transformation could be used to make this tessellation?
Solution
The regular hexagon has been
complete turn.
3 times to make a
360º
c) What other transformation could create this tessellation?
Solution
A translation can be used to make this tessellation larger.
The 3 different hexagons forming this tile can be translated
and diagonally.
( )
12.3 Constructing Tessellations Using Rotations MHR 659
Name: _____________________________________________________
Date: ______________
a) What polygons could you use to make this tessellation?
Use pattern blocks to help you.
and
b) What transformation could you use to make this tessellation?
(translations, rotations, or reflections)
c) Fill in the blanks to explain which transformation was used.
Use the words in the box to help you.
octagon
hexagon
triangle
translating
formed by
rotating
reflecting
the equilateral
vertices
sides
The white
about 1 of its
is
.
1. Kim wants to make a tessellation using a rotation.
The sum of the angles at the point of rotation must equal 360°.
Use pattern blocks
to help you.
a) Explain what happens if the sum of the angles is less than 360°.
__________________________________________________________________________
__________________________________________________________________________
b) Explain what happens if the sum of the angles is more than 360°.
__________________________________________________________________________
__________________________________________________________________________
660
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
2. Complete the table.
Tessellation
Names of Polygons
Type of Transformation
a)
b)
c)
Use pattern blocks
to help you.
3. a) Choose a polygon that you can rotate
to make a tessellation.
Draw the design.
b) Choose 2 regular polygons that you can
rotate to make a tessellation.
Draw the design.
12.3 Constructing Tessellations Using Rotations MHR 661
Name: _____________________________________________________
Date: ______________
a) Choose 1 of the pysanka designs shown.
b) Outline 1 of the designs in the pysanka with a highlighter.
c) What shapes did you highlight?
A pysanka is a
decorated egg
popular in Ukraine.
_________________________________________________________________________
d) How are the shapes tessellated?
_________________________________________________________________________
e) Make your own pysanka design by tessellating one or more polygons.
Make sure your design is big enough to fit on an egg. Colour your design.
Web Link
To see examples of pysankas,
go to www.mathlinks8.ca
and follow the links.
f) If you have time, decorate an egg with your pysanka design.
662
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
12.4 Warm Up
1. Complete the table.
Tessellation
Shape of the Tiles
Type of Transformation
a)
b)
2. Unscramble the letters to complete each sentence.
a) Tessellations can be made with 2 or more
.
GNYOLPSO
b) Two types of transformations are
and
.
RSNETFEOLIC
OSLNAASTNIRT
c) The area of a tile is the same after it is
.
MEDASRFONRT
3. The letter “L” can be used to tessellate the plane.
a) Draw a design using the letter “L” to
tessellate the plane.
b) Name another letter that can tessellate
the plane.
c) Draw a design using this letter.
12.4 Warm Up MHR 663
Name: _____________________________________________________
Date: ______________
12.4 Creating Escher-Style Tessellations
To make an Escher-style tessellation:
1. Draw an equilateral triangle with 6-cm sides.
Cut it out.
2. Inside the triangle, draw a curve that goes from 1 vertex to another on 1 side.
Cut along the curve.
3. Rotate the piece 60° counterclockwise (
) about the vertex at the top.
Tape the piece you cut off in place as shown.
This is your tile.
4. To tessellate the plane, draw around the tile on another piece of paper.
Then, rotate and draw around the tile.
Repeat this over and over to make your design.
5. Add colour and designs to the tessellation.
664
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
Working Example: Identify the Transformation Used in a Tessellation
What transformation was used to create each of the tessellations?
Tessellation B
Tessellation A
Solution
Tessellation A:
Tessellation A is made up of
that together form a
.
The transformation used to make this tessellation
is
.
Tessellation B:
Tessellation B is made up of figures that go
from white to black and then repeat.
They repeat
( ).
The transformation used to make this tessellation
is
.
What transformation was used to make this tessellation?
Explain your answer.
12.4 Creating Escher-Style Tessellations MHR 665
Name: _____________________________________________________
Date: ______________
1. Juan listed these steps to make an Escher-style tessellation.
Step
Step
Step
Step
1:
2:
3:
4:
Make sure there are no overlaps or spaces in the pattern.
Use transformations so that the pattern covers the plane.
Use a polygon.
Make sure the interior angles at the vertices total exactly 360°.
Pedro said he made a mistake.
List the steps in the correct order.
_________________
_________________
_________________
_________________
2. Complete the chart.
Tessellation
Type of Transformation(s)
a)
_________________________
b)
_________________________
c)
_________________________
_________________________
666
MHR Chapter 12: Tessellations
Shape of Tile
Name: _____________________________________________________
Date: ______________
3. a)
The original shape that was used to make this tessellation was a
.
(triangle or square)
Draw this shape on the tessellation so it has 1 complete teapot inside it.
b) Explain or show how the tessellation could have been made.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
c) Draw 1 more row on the tessellation.
12.4 Creating Escher-Style Tessellations MHR 667
Name: _____________________________________________________
Date: ______________
4. Draw an Escher-style tessellation using an equilateral triangle.
Draw an equilateral triangle on the grid.
All sides must be equal.
Add details and colour to your design.
Use translations to make an Escher-style tessellation.
5. Draw an Escher-style tessellation using squares with rotations and translations.
Draw a square on the grid.
Add details and colour to your design.
Use rotations and translations to make an Escher-style tessellation.
668
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
You are going to use an Escher-style tessellation to make a design.
This design could be used for
a binder cover
wrapping paper
a border for writing paper
a placemat
a) What will your beginning shape be?
b) Cut a simple picture out of a magazine or a comic book and use this as your shape.
or
Draw a picture to use as your shape.
c) How will you tessellate the plane?
_________________________________________________________________________
_________________________________________________________________________
d) On the grid, draw an Escher-style tessellation.
Web Link
To see examples of Escher’s
art, go to www.mathlinks8.ca
and follow the links.
12.4 Math Link MHR 669
Name: _____________________________________________________
Date: ______________
12 Chapter Review
Key Words
For #1 to #4, unscramble the letters for each puzzle.
Use the clues to help you.
Clues
1. a 2-D flat surface that
stretches in all directions
Scrambled Words
LENAP
2. using repeated shapes of
the same size to cover a
region without spaces or
overlapping
LITGIN THE
EPALN
3. a pattern that covers the
plane without overlapping
or leaving spaces
SLTIOEETANLS
4. examples are translations,
rotations, and reflections
RMTSAINFNOTAOR
Answer
12.1 Exploring Tessellations With Regular and Irregular Polygons, pages 646–651
5. Name the polygons used to make each tiling pattern.
670
a)
b)
c)
d)
MHR Chapter 12: Tessellations
Name: _____________________________________________________
Date: ______________
6. a) Explain the difference between a regular polygon and an irregular polygon.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
b) Which polygon in #5 is a regular polygon?
c) Which polygon in #5 is an irregular polygon?
12.2 Constructing Tessellations Using Translations and Reflections, pages 653–657
7. What transformation(s) could be used to make the following patterns?
a)
b)
8. Make a tiling pattern using equilateral triangles and squares.
Use 1 translation and 1 reflection to create the pattern.
Chapter Review MHR 671
Name: _____________________________________________________
Date: ______________
12.3 Constructing Tessellations Using Rotations, pages 659–662
9. What transformations could be used to make the following patterns?
a)
b)
10. Make a tessellation using this polygon.
Name the polygon that completes the pattern.
Colour it blue.
672
MHR Chapter 12: Tessellations
.
Name: _____________________________________________________
Date: ______________
12.4 Creating Escher-Style Tessellations, pages 664–669
11. This design is made up of 6 quadrilaterals.
a) How many sides does a quadrilateral have?
b) Highlight 1 of the quadrilaterals.
c) What transformation was used to make this tessellation?
12. a) Make an Escher-style tessellation.
Use only 1 shape.
b) Name the shape you used.
c) Name the transformation(s) you used.
Chapter Review MHR 673