CHAPTER
17 Preparation
Useful skills for this chapter:
• understanding of two-dimensional shapes, including the naming of attributes
Kites and darts
Sir Roger Penrose is a mathematician who uses shapes
like these kites and darts to make tiling patterns.
72°
144°
36°
216°
72°
AL
72°
PA
K
K I CF F
O
G
ES
• the ability to name angles and measure them using a protractor.
72°
36°
FI
N
Trace and cut out six
kites and six darts. Use
combinations of kites
and/or darts meeting at
one vertex to make other
interesting shapes. The
rule for joining the kites
and darts is that when two
shapes share an edge, the
red or blue patterns must
match on these edges.
There are seven shapes that you can make from kites
and darts. The first one has been done for you. Can you
find the other six?
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17
CHAPTER
Measurement and Geometry
G
ES
Symmetry and
­transformation
N
AL
PA
If you drew a line down the middle of your face, you would see that the two
halves match up exactly across the line. This is called symmetry. If you place
a small mirror along the dotted lines below, you will find that the image in the
mirror completes the picture.
FI
We see patterns all around us. Many patterns are made by shapes fitting together.
Rotation, reflection and translation are some of the different ways we can
transform a two-dimensional shape.
In this chapter we continue to discover more about the properties of two- and
three-dimensional shapes. We look at symmetry in geometry and think of how
this might apply in nature. We investigate the effect of moving two-dimensional
shapes and visualise these transformations.
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383
17A
Symmetry of twodimensional shapes
If we stand up straight, the vertical line down the centre of our face or body
divides us into two almost identical pieces. In mathematics, when the pieces of
a two-dimensional shape match up exactly across a straight line, we say the shape
is ‘symmetrical about the line’.
PA
G
ES
Draw an isosceles triangle on a piece of paper, then cut it out. Fold the right-hand side
of the triangle over so it lies exactly on the left-hand side of the triangle and make a
crease down the centre of the triangle. The halves of the triangle on either side of the
fold line should match exactly. The fold line is called the axis of symmetry or the line
of symmetry. Use a protractor to measure the angles where the fold line meets the
base. They should be 90°.
AL
The isosceles triangle is symmetrical about the fold line.
FI
N
Some shapes have more than one line of symmetry.
These shapes have two lines of symmetry.
This shape has three lines of symmetry.
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17A
Individual
a
b
c
d
G
ES
1 How many lines of symmetry does each quadrilateral have?
2 a Draw 5 regular polygons of different sizes.
b Mark dotted lines on the polygons to show all the lines of symmetry.
PA
3 Write the alphabet in capital letters. List the letters that have at least one line
of symmetry.
4 a Draw a right-angled triangle with a short side equal to 5 cm and a long
side perpendicular to it equal to 7 cm. Mark in the right angle and draw in
the third side.
AL
b Draw the shape you would get if the 7 cm line is the axis of symmetry for
a new shape and your triangle is one half of it.
c Draw the shape you would get if the 5 cm line is the axis of symmetry for
a new shape.
5 This shape is made from 4 identical small squares.
FI
N
It has 4 lines of symmetry.
Use 4 identical squares to make a shape that has:
a 1 line of symmetry
b 2 lines of symmetry
Use 5 identical squares to make a shape that has:
c 1 line of symmetry
d 4 lines of symmetry
CHAPTER 17 e 0 lines of symmetry
S ymmetry a n d ­tra n s f o rmati o n
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385
17B
Transformation
and tessellation
Rotation, reflection and translation are some of the different ways we can transform a
two-dimensional shape.
Rotation
We rotate a shape when we turn it through
an angle.
image
G
ES
The diagram at the right shows an arrow shape
pointing upwards. The shape has been rotated
clockwise around the red dot three times, each
time by 90°. The word ‘image’ has been used to
label the shape in each new position.
image
PA
image
We can rotate clockwise or anticlockwise about a point.
FI
N
O
image
AL
This triangle has been turned 90°
in a clockwise direction about O.
image
This triangle has been rotated 90° anticlockwise.
O
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Example 1
image
How has this shape been moved?
O
The rectangle has been rotated 90° in
a clockwise direction.
image
G
ES
Solution
Reflection
PA
O
FI
N
AL
A reflection is a transformation that flips a figure about a line. This line is called the
axis of reflection. A good way to understand this is to suppose that you have a book
with clear plastic pages and a triangle drawn on one page, as in the first diagram
below. If the page is turned, the triangle is flipped over. We say it has been reflected;
in this case the axis of reflection is the binding of the book.
This shape has been reflected in the vertical line.
egami
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387
Example 2
Which transformation has been used to
produce the image?
egami
Solution
G
ES
The image is the mirror image of the shape on the left. It has been reflected in
the vertical line.
Translation
When we translate a shape, we slide it. We can slide it left or right, up or down.
Transformations move the shape without rotating it.
PA
This shape has been
translated horizontally.
This shape has been
translated vertically.
image
AL
image
Example 3
image
FI
N
How has this shape been moved?
Solution
The shape has been translated horizontally.
Tessellation
A tessellation is a tiling pattern made by fitting together transformations of a
two-dimensional shape with no gaps or overlaps. The tessellation can continue in
all directions.
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Start with an equilateral triangle.
G
ES
We can rotate it 180° and translate it so the triangles fit
together perfectly. The tiling goes on forever. We say
that the equilateral triangle tessellates.
AL
PA
The shape used in the pattern below is not a tessellating shape because we cannot
rotate and translate it to fill up the whole space without gaps or overlaps.
Example 4
N
Will this shape tessellate?
FI
Solution
Yes, this shape will tessellate. It can be rotated 180° and translated so the pieces fit
together without any gaps or overlaps.
The pattern can be continued horizontally and vertically as far as you wish.
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389
17 B Whole class
CONNECT, APPLY AND BUILD
1 Describe the transformations.
G
ES
b
a
image
PA
image
c
d
image
FI
e
N
AL
image
f
image
image
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2 Look around the school for tessellating patterns such as brickwork, floor tiles,
wall tiles, wallpaper, carpets and mats. Draw a sketch of three interesting
tessellating patterns.
3 a Trace a hexagon from your class set of shapes.
b Measure the internal angles of the hexagon and label your drawing.
G
ES
c Draw two more hexagons in a tessellating pattern as shown below and mark
in the size of the angles A, B and C.
B
C
PA
A
d What is the sum of the angles A, B and C?
e What does this tell you about tessellating patterns? Complete this sentence:
AL
In a tessellation the angles about a point sum to ________ degrees.
4 Use attribute blocks or pattern blocks.
a Choose a shape (not a hexagon) that you think will tessellate. Show that the
shape tessellates by putting together at least 10 tracings of the shape.
N
b Measure or calculate the angles about a point within your tessellating pattern.
FI
5 Use blocks to make a tiling pattern with two or more different-shaped tiles.
Which shapes did you use? Measure the angles about a point within your
tessellating pattern.
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391
17 B
Individual
1 Describe the transformation of these shapes.
b
a
c
image
image
G
ES
image
2 Draw these shapes, then draw what they look like after they have been reflected
in the line.
b
PA
a
3 Draw these shapes, then draw what they look like after they have been translated.
b Translate vertically
AL
a Translate horizontally
N
4 For each of these shapes:
• rotate the shape 90º clockwise
• draw the image
FI
• repeat the above steps twice.
a
b
c
5 Select one shape from your class set of shapes that will tessellate. Draw a
tessellation using the shape. Colour the shapes to show the pattern you have
made.
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