CHAPTER Tessellations

Transcription

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
ΔABC ≅ ΔDEF
∠A means angle A.
AB means line segment AB.
∠A = ∠D
AB = DE
∠B = ∠E
AC = DF
∠C = ∠F
A
E
+
º
D
B
BC = EF
º
º
+
º
C
F
1. Are the figures in each pair congruent? Circle the correct answer.
a)
P
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
ΔABC has been translated 4 units
vertically ( b ).
The translation image is ΔA′B′C′.
●
y
4 P’
y
A
C
–2A’ 0 x
–4
0 S
–2
B’
Q’
S’
R’
2 line of reflection
m
2
B
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.
2 R4 x
–2 P
C’
●
Q
rotations—also called turns
ΔDEF has been rotated
180º counterclockwise
around the origin.
ΔD′E′F′ is the rotation image.
y
centre of
rotation
2
(2, –2),
, and
F
E9
2 x
–2
y
1
.
ΔTHE has been rotated
F9
0
D
E
3. ΔTHE is rotated around the centre of rotation, z.
The coordinates of ΔT′H′E′ are
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 ΔMON 3 units up and
4 units left.
b) Use the x-axis as the line of reflection to reflect ΔMON.
Use prime notation.
y
4
3
2
1
–4 –3 –2 –1 0
–1
2
1
3
M
4 x
O
–2
–3
N
–4
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 ΔABC congruent to ΔXYZ? Circle YES or NO.
Give 1 reason for your answer.
_________________________________________________________________________
d) Are ΔABC and ΔXYZ regular or irregular? Circle REGULAR or IRREGULAR.
_________________________________________________________________________
Copy ΔABC or ΔXYZ 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.
e)

644
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°.
be
°.
°.
Name: _____________________________________________________
Date: ______________
Which of the shapes can be used to tessellate the plane?
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.
Arrange 3 shapes along a side with
the same length.
Draw the diagram.
120° 60°
60° 120°
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)
c)
50º
70º
60º
(can or cannot)
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.
+
+
=
Is this equal to a full turn? Circle YES or NO.
The polygon
tessellate the plane.
(can or cannot)
648
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
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.
a)
y
6
A
A’
• A translation is a slide.
B
D
4
B’
2
C
–2
• A rotation is a turn.
D’
• A reflection is a mirror image.
C’
0
2
4
x
n
b)
c)
y
6
y
0
4
–2
2
–4
E
R
–6
–4
–2
0
2
4
6 x
–2
2. Write the names of the polygons used in each tessellation.
a)
652
b)
8 x
6
4
2
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
.
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.
_____________________________________________________________________________
_____________________________________________________________________________
654
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
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.
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
y
y
2
2
4
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
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?
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
Name: _____________________________________________________
Date: ______________
2. Complete the table.
Tessellation
Names of Polygons
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
Name: _____________________________________________________
Date: ______________
12.4 Warm Up
1. Complete the table.
Tessellation
Shape of the Tiles
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
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
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
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.
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)
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
.
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
Name: _____________________________________________________
Date: ______________
12 Practice Test
For #1 to #4, circle the best answer.
1. Which regular polygon cannot be used to tile a plane?
A square
B triangle
C hexagon
D pentagon
2. Polygons can be used to make a tessellation.
The interior angles must add up to
° where the vertices of the polygons meet.
A 90°
B 180°
C 270°
D 360°
3. Which polygon can be used to make a tessellation?
A regular pentagon
B regular hexagon
C regular heptagon
D regular octagon
4. How many different polygons were used to make this design?
674
A 1
B 2
C 3
D 4
Name: _____________________________________________________
Date: ______________
Short Answer
5. Decide if each statement is true or false.
Circle TRUE or FALSE.
If the statement is false, rewrite it to make it true.
a) Tessellations need more than 2 polygons to make a design. TRUE or FALSE
_________________________________________________________________________
_________________________________________________________________________
b) Tessellations can be made if the interior angles of the polygons equal exactly 360° where
the polygons meet. TRUE or FALSE
_________________________________________________________________________
_________________________________________________________________________
c) Rotations cannot be used to make tessellations. TRUE or FALSE
_________________________________________________________________________
_________________________________________________________________________
6. Can Jamie make a tessellation using this triangle? Circle YES or NO.
Explain your answer.
50º
20º
110º
____________________________________________________________________________
____________________________________________________________________________
7. a) What type of polygon is used to make this design?
Circle PARALLELOGRAM or RECTANGLE or QUADRILATERAL.
b) What transformation is used in the pattern?
Practice Test ● MHR 675
Name: _____________________________________________________
Date: ______________
8. Describe how you would make this tessellation.
_______________________________________
_______________________________________
_______________________________________
_______________________________________
9. Make an Escher-style tessellation using an equilateral triangle or a square.
You are going to make a mosaic design to hang in your room.
You must include at least 2 different shapes and 1 transformation.
a) Shape 1:
Shape 2:
b) Type of transformation:
c) List the materials you will need to make your pattern.
Examples: coloured
construction paper, coloured
transparencies, tile pieces,
grid paper, paints.
d) Make your mosaic design on a separate sheet of paper.
e) After you have completed your design, write a short paragraph about it on a separate sheet
of paper.
• Describe the different shapes and transformations you used to make your mosaic.
• Explain why you chose the shapes, transformation, materials, and colours that you used.
676
Name: _____________________________________________________
Date: ______________
Use the clues to find the key words from Chapter 12.
Write them in the crossword puzzle.
Across
4. A figure with many sides
6. A pattern that covers an area without overlapping or leaving spaces
8. Examples are reflections, rotations, and translations
9. A 2-dimensional flat surface that stretches in all directions
Down
1. A figure with 3 sides
7. The name of the artist who used tessellations to make different pieces of art
1
2
3
5
4
6
7
8
9
Key Word Builder ● MHR
677
Name: _____________________________________________________
Date: ______________
Math Games
Playing at Tiling
Game boards can be made from polygons that tessellate.
For example, chessboards are made from squares.
This board includes squares and regular octagons.
●
●
49
50
41
39
40
31
30
20
11
1
23
12
9
2
25
16
17
8
3
26
27
14
13
●
36
35
24
18
19
10
34
28
29
22
45
37
38
33
32
21
44
43
42
46
47
48
1 Playing at Tiling game
board for each small
group
two 6-sided dice for each
small group
1 coloured counter for
each student
15
6
7
4
5
1. Play a game on this board with a partner or in a small group.
Rules:
● Each player rolls a die to see who plays first.
The highest roll goes first. If there is a tie, roll again.
● For each turn, roll the 2 dice. Use the greater number.
●
●
●
●
Starting at #1, move your counter that number
of spaces ahead.
If you roll a double, move to the next space
that is a different shape from the shape you’re on.
Then move ahead the number spaces equal to the
value on 1 of the die.
The first player to reach 50 wins.
2. Design your own game board.
● Use 1 or 2 shapes.
● Your shapes must tessellate your board.
I rolled a 3 and a 5, so I
move 5 spots ahead.
I rolled two 4s when my counter
was on square 13. I move ahead to
the next octagon, number 16. Then I
move ahead 4 spaces to 20.
Use pattern blocks to
help you find a shape to
tessellate.
The board will have no spaces or shapes that overlap.
● On a separate piece of paper, write the rules for a dice game to be played on your board.
● Play your game with a partner.
● Change any of your rules to make your game better.
678
Name: _____________________________________________________
Date: ______________
Challenge in Real Life
Border Design
Designers make patterns and border designs for tiles, wallpaper, fabrics,
and rugs.
Design a border for the wall at the skateboard park.
Use what you know about tessellations to make your design for a border.
1. On construction paper, draw and cut out an equilateral
triangle or a square. This is your template.
2. Use your template to make transformations of your shape.
Draw a sketch of each transformation.
Reflection:
Rotation:
●
●
●
●
construction paper
scissors
coloured pencils or
markers
grid paper
Templates are used to
trace the same shape on
designs.
Translation:
3. Design your border on a piece of grid paper that is 12 cm × 28 cm.
Use at least 1 of your transformations to make your border.
4. Colour your border to show your transformations.
Challenge in Real Life ● MHR 679
Name: _____________________________________________________
Date: ______________
Chapters 9—12 Review
Chapter 9 Linear Relations
1. a) The table of values shows the number of triangles in an increasing pattern. Complete the table.
1
3
Figure Number, f
Number of Triangles, n
2
5
b) Graph the table of values.
3
7
4
c) Does your graph represent a linear
relation? Circle YES or NO.
n
10
________________________________
8
6
________________________________
4
2
0
1
2
3
4
________________________________
f
2. The graph is a linear relation.
It shows the amount you pay for an item in relation to how many items you buy.
C
a) What is the cost if you buy 1 item?
18
16
b) Complete these statements:
for 1 item.
The cost increases by $
every time you buy
another item.
To move from 1 point to the next, move 1 unit horizontally
12
Cost ($)
The cost starts at $
14
10
8
6
4
2
(↔) and
units vertically (↕).
c) Complete a table of values for this linear relation.
Quantity, n
Cost, C ($)
d) What is an expression for the cost in terms of the quantity?
e) If the quantity is 8, what is the cost?
680
0
1
2
3
4
Quantity
5
6 t
Name: _____________________________________________________
Date: ______________
3. A farmer is building a post-and-rail fence around his yard.
The formula r = 3p – 3 represents the number of rails in relation to the number of posts,
where r is the number of rails and p is the number of posts.
rail
post
1 section
3 sections
2 sections
4 sections
a) Draw the next 2 pictures of the fence showing 3 sections and 4 sections.
b) Complete the table of values. Use the drawing to help you.
Number of Posts (p)
2
3
4
5
6
7
Number of Rails (r)
c) Graph the table of values.
20
18
To draw a graph:
Label each of the axes using p and r.
Describe each axis.
Give the graph a title.
Plot the points.
16
14
12
10
8
6
d) Does the relation appear to be linear?
Circle YES or NO.
4
2
O
___________________________________________
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5 x
___________________________________________
4. a) Complete the table of values using 4 positive
integer values and 4 negative integer values.
y
y = 2x – 3
x
–4
b) Graph the relation.
12
y
Find the y-value for x = 0:
y = 2x – 3
y = 2(0) – 3
10
8
6
4
2
y=
0
y=
–3
-5 -4 -3 -2 -1 O
-2
-4
-6
-8
-10
-12
Chapters 9–12 Review ● MHR 681
Name: _____________________________________________________
Date: ______________
Chapter 10 Solving Linear Equations
5. a) What equation does this diagram show?
x
x
x
x
b) Solve the equation.
6. Use models or diagrams to solve each equation.
a)
s
= –5
2
Check:
Left Side
b) 2(x – 5) = –4
Right Side
Check:
Left Side
Right Side
7. Solve each equation. Check your answers.
a)
x
= –4
7
Check:
Left Side
682
b) 5x – 26 = 14
Right Side
Check:
Left Side
Right Side
Name: _____________________________________________________ Date: ______________
8. Jason’s age is 3 years less than
1
of his father’s age.
3
a) Write an expression for Jason’s age.
Use f to represent his father’s age.
b) If Jason is 10 years old, how old is his
father?
Equation →
f − __________ = Jason’s age
Solve →
9. Elijah works for a diamond mine. He is paid r dollars per hour.
When he works the late shift, $2 is added to his regular hourly rate.
a) What expression represents his hourly rate for the late shift?
b) He works the late shift for 6 h.
The expression 6(r + 2) shows how much he would make.
What is the expression if he worked the late shift for 40 h?
c) Elijah made $960 after working the late shift for 40 h.
Write an equation for this problem.
d) Solve the equation to find how much he makes per hour.
Elijah makes
per hour.
e) How much does Elijah make per hour for working the late shift?
Sentence: _________________________________________________________________
Name: _____________________________________________________
Date: ______________
Chapter 11 Probability
10. Use the spinner to answer the questions.
1
Probability =
0
2
4
favourable outcomes
possible outcomes
3
a) What is the probability of spinning an odd number?
P(
Odd numbers are 1, 3, 5 …
)=
b) What is the probability of spinning an even number?
Even numbers are 0, 2, 4, 6 …
P(
)=
c) If you spin the spinner twice, what is P(odd number, then even number)?
P(odd number, then even number) = P(odd number) × P(even number)
×
=
=
11. A computer store has a sale.
You can buy 1 of 4 different computers, and 1 of 3 different printers.
How many combinations are there?
Total possible outcomes = number of different computers × number of different printers
=
×
=
Sentence: __________________________________________________________________
684
Name: _____________________________________________________
Date: ______________
12. Gillian flips a disk labelled T on 1 side and H on the other.
She spins a spinner that is divided into 3 equal sections labelled T, H, and O.
a) What is the probability of flipping an H on the disk?
P(
H
H
)=
O
T
b) What is the probability of spinning an H on the spinner?
P(
)=
c) What is the probability there will be an H on both? Complete the table to find your answer.
Spinner
O
H
Disk
T
H
T
P(H, H) =
=
← decimal
=
← percent
d) Use multiplication to check your answer to part c).
P(H on disk, H on spinner) = P(H on disk) × P(H on spinner)
=
×
=
Name: _____________________________________________________
Date: ______________
13. In every box of cereal you have the chance of getting a flying disk
that is red, blue, yellow, or green.
a) Conduct a simulation using the spinner to find the colour of the
disks in the next 2 boxes of cereal.
Trials
Example
Spin 1
Green (G)
Spin 2
Yellow (Y)
Red
Green
Blue
Yellow
Result
G, Y
1
2
3
4
5
6
7
8
9
10
b) What is the experimental probability that the next 2 boxes of cereal will each have a blue
disk in them?
Check your results
from the chart above.
P(both blue) =
c) What is the theoretical probability that the next 2 boxes of cereal will have blue disks
in them?
P(both blue) = P(B) × P(B)
=
4
×
=
686
4
Name: _____________________________________________________
Date: ______________
Chapter 12 Tessellations
14. Polygons are used to tile the plane.
95º
90º
90º
90º
a) The squares have been formed into a tessellation.
Show how you know.
90º
The sum of the interior angles where the vertices of the polygon
meet equals
.
90º
90º 90º
90º
+
+
+
=
This is a full turn.
b) Will the pentagon tessellate the plane? Circle YES or NO.
Show how you know.
95º
95ºº 95º
95º
15. a) Make a tessellation.
Use a square and 1 other shape.
b) Describe the transformation(s) you used
for your pattern.
________________________________
________________________________
________________________________
________________________________
16. What transformation was used in this design?
Name: _____________________________________________________
Date: ______________
Put Out a Forest Fire
●
●
●
●
One way to fight a forest fire is to drop water and fire retardant
on it from an airplane.
You are training to be a firefighting airplane pilot.
Create a simulation to see how effective you are at putting out a fire.
Triangle to Tessellate
BLM
ruler
coloured pencils (orange,
green, blue)
modelling clay or bingo
chips
Fire retardant is a
chemical that helps put
out and prevent fires.
1. Draw a 14 cm by 16 cm rectangle on a blank sheet of paper.
Cut out the triangle from the Triangle to Tessellate BLM.
The full triangle counts as 2 shapes. Half of the triangle counts as 1 shape.
1 Shape
2 Shape
Using transformations and your triangle, tile your paper until the rectangle is full.
Use full and half triangles to completely cover the rectangle.
Colour the shapes in your tessellation using the ratio 1 blue : 3 orange : 4 green.
Cut out the rectangle.
Join your tessellated rectangle with 3 other students’ rectangles.
This large tessellation makes a map of a forest fire.
● Orange shows the area that is burning.
● Green is the forest.
● Blue is the lakes.
688
Name: _____________________________________________________
Date: ______________
2. Try to put out the fire by dropping “water” on each orange area.
One at a time, stand beside the map and drop 3 pieces of modelling clay onto it.
Each drop represents a planeload of water.
Record what colour each load of water lands on.
Orange
Green
Blue
3. a) Hitting an orange shape puts out the fire in that part and all the orange shapes attached to it.
Find the experimental probability of hitting an orange shape.
Experimental Probability =
number of orange half triangles where fire was put out
total number of all half triangles
=
=
← decimal
=
← percent
b) Find the theoretical probability of hitting an orange shape.
Look at the total tessellated map.
Theoretical Probability =
number of orange half triangles
total number of all half triangles
=
=
← decimal
=
← percent
Task ● MHR 689
Answers
6. Answers will vary. Example:
Get Ready, pages 642–643
1. a) not congruent
b) congruent
2. a) regular b) irregular
3. (0, –2); (0, 0); counterclockwise or clockwise
4. a) and b)
y
4
3
O’
2
1
translation
M’
N’
–4 –3 –2 –1 0
–1
–2
–3
–4
reflection
N’
O’
M’
1 2 3 4 x
O
M
N
Math Link
a)
AC
= 35 mm,
AB
= 25 mm,
CB
= 33 mm,
ZX
= 35 mm,
XY
= 25 mm,
ZY
= 33 mm
Math Link
a) Answers may vary. Example:
Shape
1
2
3
4
b) ∠A = 63°, ∠B = 70°, ∠C = 47°, ∠X = 63°, ∠Y = 70°, ∠Z = 47°
c) YES. They are congruent because all angles and sides correspond.
d) IRREGULAR. They are irregular because not all the angles are equal.
e) Answers will vary.
12.1 Warm Up, page 645
b) and c) Answers will vary.
1. a) octagon b) square c) equilateral triangle d) isosceles triangle
e) pentagon f) hexagon
12.2 Warm Up, pages 652
2. a) ∠A = 108°, ∠B = 108°, ∠C = 108°, ∠D = 108°, ∠E = 108°;
AB = 2 cm, BC = 2 cm, CD = 2 cm, DE = 2 cm, AE = 2 cm
2. a) square, triangle b) squares, octagons
b) Answers may vary. Example: All the sides are the same length, and all
the angles are equal. c) regular pentagon
12.1 Exploring Tessellations With Regular and Irregular
Polygons, pages 646–651
Working Example: Show You Know
Regular Polygon?
Yes/No
no
no
yes
yes
Name of Shape
octagon
hexagon
small square
large square
1. a) reflection b) translation c) rotation
12.2 Constructing Tessellations Using Translations and
Reflections, pages 653–657
squares, triangles; translations
Communicate the Ideas
a) can b) can c) will tessellate; parallelogram
1. BRENT. If it were just reflecting, the polygon it would continue in a
straight line, and would not make the same pattern that is shown.
Practise
1. a) Answers will vary. Example:
2. a) regular hexagon, equilateral triangle; translation or reflection
b) square, equilateral triangle; reflection
c) parallelogram, triangle; translation and reflection
Apply
square
b) Answers will vary. Example: Each angle measures 90°.
c) Answers will vary. Example: The sum of the interior angles is 360°
where the vertices meet.
3. Answers may vary. Example:
Practise
2. can
b) Answers will vary. Example:
4. Answers will vary. Example: square tiles on floors, rectangular bricks
on walls.
Apply
690
4. a) 360° b) Answers may vary. Example:
c) The sum of the interior angle measures at the point where the vertices
of the brick meet is 360°.
b) hexagon, triangle
c) Answers will vary.
Math Link
a) triangles, squares b) square c) vertically, horizontally
d) Answers will vary.
b) Answers will vary. Example: T c) Answers will vary. Example:
12.3 Warm Up, page 658
1. a)
b)
c)
12.4 Creating Escher-Style Tessellations, pages 664–669
A rotation, because the same shape has been rotated to form the tessellation.
d)
e)
2.
f)
1. Step 1: Use a polygon. Step 2: Make sure there are no overlaps or gaps in
the pattern. Step 3: Make sure the interior angles at the vertices total
exactly 360°. Step 4: Use transformations so that the pattern covers the
plane.
y
Practise
2. a) translation; parallelogram b) rotation; triangle c) rotation, reflection;
parallelogram
2
0
–2
2 x
–2
Apply
3. a) square b) Answers may vary. Example: The shape was cut to make the
shape of a teapot. Parts of the square were cut off from one side and
attached to another part. No part of the square was removed.
c)
3. a) parallelograms b) parallelograms, triangles
4. a) translation b) rotation c) reflection
12.3 Constructing Tessellations Using Rotations, pages 659–662
a) hexagon, triangle b) rotations c) hexagon, rotating, triangle, vertices
1. a) Answers may vary. Example: If the sum of the angles is less than 360°,
there will be gaps.
b) Answers may vary. Example: If the sum of the angles is more than
360°, the shapes will overlap.
Practise
2. a) square; rotation b) regular octagon and triangle; rotation and
translation c) cross shape and square; rotation and translation
Apply
Math Link
a)–d) Answers will vary.
Chapter Review, pages 670–673
1. plane 2. tiling the plane 3. tessellation 4. transformation
5. a) regular hexagon, equilateral triangle b) rhombus, isosceles triangle,
regular hexagon c) regular hexagon, equilateral triangle d) regular
hexagon, parallelogram, equilateral triangle
Math Link
a)–f) Answers will vary.
6. a) Answers may vary. Example: Regular polygons have equal interior
angle measures and equal side lengths; irregular polygons do not.
b) regular hexagon; equilateral triangle c) isosceles triangle; rhombus;
parallelogram
12.4 Warm Up, pages 663
7. a) rotation b) reflection, translation
1. a) regular hexagon, triangle; rotation b) regular hexagon triangle;
rotation, reflection, translation
2. a) polygons b) translations, reflections c) transformed
3. a) Answers may vary. Example:
9. a) rotation b) reflection, rotation, translation
Answers ● MHR 691
d) C = 3n e) 24
10. square; Answers may vary. Example:
3. a)
rail
post
one section
11. a) 4 b) Answers may vary. c) rotation
three sections
two sections
four sections
b)
2
3
Number of Posts (p)
Number of Rails (r)
b) Answers will vary.
c)
3
6
4
9
5
12
6
15
7
18
d) YES. The points lie in a
straight line.
y
20
18
16
14
12
10
Practice Test, pages 674–676
1. D 2. D 3. B 4. B
5. a) FALSE Answers may vary. Example: Tessellations can be made with
1 polygon. b) TRUE c) FALSE. Answers may vary. Example:
Rotations can be used to make tessellations.
6. YES. Answers may vary. Example: Any triangle can create a tessellation.
Two congruent triangles form a parallelogram that tiles the plane.
7. a) QUADRILATERAL b) rotation
8
6
4
2
0
1
2
3 4
7
8
9 10 x
4. Answers may vary. Example:
a) y = 2x – 3
8. Answers may vary. Example: Rotate the top left pentagon about the centre
of the black square for a full turn to form a combined shape of 4
pentagons with the square at the centre. Translate this combined shape to
create the tessellation.
5 6
x
–4
–3
–2
–1
0
1
2
3
4
b)
y
–11
–9
–7
–5
–3
–1
1
3
5
y
8
4
−4
−2 0
–4
2
4
x
–8
–12
Wrap It Up!, page 676
5. a) 4x = 12 b) x = 3
a)–e) Answers will vary.
6. a) s = –10 b) x = 3
Key Word Builder, page 677
7. a) x = –28 b) x = 8
1
8. a) f – 3 b) 39 years old
3
9. a) r + 2 b) 40(r + 2) c) 40(r + 2) = 960 d) $22.00 e) $24.00
2
3
6
10. a)
b)
c)
5
5
25
11. There are 12 combinations of computers and printers.
1
1
1
1
12. a) P(H on disk) =
b) P(H on spinner) =
c) P(H, H) =
d)
2
3
6
6
1
13. a) and b) Answers will vary. c)
16
14. a) 90° + 90° + 90° + 90° = 360° b) NO. The interior angles add up to
380°, which is more than a full turn.
Across
4. polygon 6. tessellation 8. transformation 9. plane
Down
1. triangle 2. hexagon 3. quadrilateral 5. octagon 7. Escher
Challenge in Real Life, page 679
Answers will vary. Example:
1.
2. Answers will vary.
3.
Chapters 9–12 Review, pages 680–687
c) YES. The points lie in a straight line.
1. a) 9 b) 10n
8
6
4
2
0
1
2
3 4
f
2. a) $3.00 b) $3.00; $3.00; 3
16. translation
c)
Task, page 688
Quantity, n
Cost, C ($)
692
1
3
2
6
3
9
4
12
5
15
6
18
Answer will vary.
b) Answers will vary.
Example: rotation and translation