transformations - mrs. purtle's math classes · reflections and line symmetry rotations and...

Transformations

22Chapter

Contents: A

B

C

D

E

Reflections and line symmetry

Rotations and rotationalsymmetry

Translations

Enlargements and reductions

Tessellations

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y:\HAESE\IB_MYP1\IB_MYP1_22\403IB_MYP1_22.CDR Thursday, 10 July 2008 9:07:37 AM PETER

OPENING PROBLEM

TRANSFORMATIONS

Translation, reflection, rotation, and enlargement are all transformations.

For example:

a translation

a rotation about O

a reflection

mirror line

an enlargement

When we perform a transformation, the original shape is called the object. The shape which

results from the transformation is called the image.

Consider an equilateral triangle.

Things to think about:

²

² Can you find the centre of rotation of an equilateral triangle? The figure must rotate

about this point and fit exactly onto itself in less than one full turn.

How many times would an equilateral triangle fit onto itself in one full turn?

² Make a pattern using equilateral triangles so there are no gaps between the triangles

and the edges meet exactly.

Can you draw a mirror line on an equilateraltriangle? The figure must fold onto itself alongthat line, so it matches exactly.

How many of these mirror lines can you draw?

shift a particular distance in a particular direction

O

404 TRANSFORMATIONS (Chapter 22)

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ACTIVITY 1 TRANSFORMING CATS

CONGRUENT FIGURES

Two figures are congruent if they have exactly the same

size and shape.

If one figure is cut out and it can be placed exactly on top

of the other, then these figures are congruent.

The image and the object for a translation, rotation, or

reflection are always congruent. The image and the object

for an enlargement are not congruent because they are not

the same size.

Following is a fabric pattern which features cats and pairs of cats. The rows

and columns have been numbered to identify each individual picture.

What to do:

1 Start with row 1 and column 1 cat.

a Give the row and column numbers for translations of this cat.

b Give the row and column numbers for rotations of this cat.

c Give the row number in column 1 for an enlargement of this cat.

d Give the row numbers in column 3 for cats congruent to this cat.

2 Start with row 1 column 2 cat.

a Give the row and column numbers for translations of this cat.

b Discuss why row 4 column 2 is not a rotation of this cat.

c Give the row and column numbers for rotations of this cat.

d Give the row number in column 1 for a reflection of this cat.

We are congruent.

Row 1

Row 2

Row 3

Row 4

Row 5

Col. 6Col. 5Col. 4Col. 3Col. 2Col. 1

TRANSFORMATIONS (Chapter 22) 405

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Y:\HAESE\IB_MYP1\IB_MYP1_22\405IB_MYP1_22.CDR Friday, 11 July 2008 4:24:13 PM PETER

3 Start with row 4 column 5 cat.

a Give the row numbers for any column 2cats congruent to this one.

b Give the row and column numbers of any

cats that are a rotation of this cat.

c Give the row and column numbers for any

cats that are a reduction of this cat.

4

5 Which two transformations are used to move the cats in

a row 1 column 1 to row 1 column 5 b row 2 column 4 to row 5 column 4

c row 2 column 5 to row 5 column 5 d row 3 column 6 to row 2 column 6?

REFLECTIONS

To reflect an object in a mirror line we

draw lines at right angles to the mirror

line which pass through key points on the

object. The images of these points are the

same distance away from the mirror line as the object

points, but on the opposite side of the mirror line.

Draw the mirror image of:

EXERCISE 22A.1

1 Place a mirror on the mirror line shown using dashes, and observe the mirror image.

Draw the object and its mirror image in your work book.

a b c

REFLECTIONS AND LINE SYMMETRYA

Example 1 Self Tutor

A is theopposite of an

enlargement. It makesthe figure smaller.

reduction

DEMO

object image

mirror line

mirror

object

image

SHAM

What is the transformation shown in the pair of cats?


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2 a Draw the image of the following if a mirror was placed on the mirror line shown:

i ii iii

b Check your answers to a using a mirror.

3 On grid paper, reflect the geometrical shape in the mirror line shown:

a b c

LINE SYMMETRY

A line of symmetry is a line along which a shape may be folded so that both parts of the

shape will match.

For example:

If a mirror is placed along the line of symmetry, the reflection in the mirror will be exactly

the same as the half of the figure “behind” the mirror.

A shape has line symmetry if it has at least one line of symmetry.

For each of the following figures, draw all lines of symmetry:

a b c


PRINTABLE

DIAGRAMS

line of symmetryormirror line

fold line DEMO


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a

4 lines of symmetry

b

no lines of symmetry

c

3 lines of symmetry

EXERCISE 22A.2

1 Copy the following figures and draw the lines of symmetry. Check your answers using

a mirror.

a b c

d e f

g h i

2 a Copy the following shapes and draw in all lines of symmetry.

i ii iii iv

b Which of these figures has the most lines of symmetry?

3 How many lines of symmetry do these patterns have?

a b

4 a How many lines of symmetry can a triangle have? Draw all possible cases.

b How many lines of symmetry can a quadrilateral have? Draw all possible cases.


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ACTIVITY 2 MAKING SYMMETRICAL SHAPES

You will need: paper, scissors, pencil, ink or paint

1 Take a piece of paper and fold it in

half.

2 Cut out a shape along the fold line.

3 Open out the sheet of paper and observe the shapes

revealed.

4 Record any observations about symmetry that you

notice.

5 Try the following:

a Fold the paper twice before cutting out your shape.

b Fold the paper three times before cutting out your shape.

In each case record your observations about the number of lines of symmetry.

6 Place a blob of ink or paint in the centre of a rectangular sheet of paper. Fold the

paper in half and press the two pieces together. Open the paper and comment on the

symmetry observed.

7

We are all familiar with things that rotate, such as the hands on a clock or the wheels of a

motorbike.

B ROTATIONS ANDROTATIONAL SYMMETRY

inkfold

fold line

ink blot

DEMO


Make symmetrical patterns by folding a piece ofpaper a number of times and cutting out a shape.How many folds would you need and what shapewould you need to cut out to get the result shown?

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The point about which the hands of the clock, or the spokes of the wheel rotate, is called the

centre of rotation.

The angle through which the hands or the spokes turn is called the angle of rotation.

The globe of the world rotates about a line called the axis of rotation.

During a rotation, the distance of any point from the centre of rotation does not change.

A rotation is the turning of a shape or figure about a point and through a

given angle.

For example:

The figure is rotated

anticlockwise about O

through 90o.



through 180o.



through 90o.

You will notice that under a rotation, the figure does not change in size or shape.

In mathematics we rotate in an anticlockwise direction unless we are told otherwise.

You should remember that 90o is a 14

-turn, 180o is a 12

-turn, 270o is a 34

-turn, and

360o is a full turn.

Rotate the given figures about O through the angle indicated:

a b c

a b c


DEMO

90°

O

180°O

90°

O

O

180°

O

90°

O

270°

O

180°

O

90°

O

270°


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EXERCISE 22B.1

1 Consider the rotations of which follow:

A B C D

Which of A, B, C, or D is a rotation of the object through:

a 180o b 360o c 90o d 270o?

2 Copy and rotate each of the following shapes about the centre of rotation O, for the

number of degrees shown. You could use tracing paper to help you.

a b c

d e f

g h i

3 Rotate about O through the angle given:

a 90o b 180o c 270o

d 180o e 90o f 270o

O

O

O

O

O

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DEMO

90°

O

180°O 270°O

O 360° 270°

O

90°

O

180°

O

90° 90°O

OO

O

OO O


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ROTATIONAL SYMMETRY

This fabric pattern also shows rotational symmetry.

If a figure has more than one line of symmetry then it will also have rotational symmetry.

The centre of rotational symmetry will be the point where the lines of symmetry meet.

For the following figures, find the centre of rotational symmetry.

a b c

a

centre is O

b

centre is O

c

centre is O

THE ORDER OF ROTATIONAL SYMMETRY


O

A full rotation does notmean that a shape hasrotational symmetry.

Every shape fits exactlyonto itself after arotation of .360o

A shape has

if it can be fitted onto itself by

turning it through an angle of

, or one full turn.

Theis the point about

which a shape can be rotated ontoitself.

The ‘windmill’ shown will fit onto itself every time it is

turned about O through . O is the centre of

rotational symmetry.

rotational symmetry

less

than

centre of rotational

symmetry

360o

90o

DEMO

O

O

O

DEMO

The is the number of times a figuremaps onto itself during one complete turn about the centre.

order of rotational symmetry


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ACTIVITY 3 USING TECHNOLOGY TO ROTATE

For example:

EXERCISE 22B.2

1 For each of the following shapes, find the centre of rotational symmetry:

a b c

2 For each of the following shapes find the order of rotational symmetry. You may use

tracing paper to help you.

a b c d

e f g h

3 Design your own shape which has order of rotational symmetry of:

a 2 b 3 c 4 d 6

In this activity we use a computer package to construct a shape

that has rotational symmetry.

What to do:

1 Click on the icon to load the software.

2 From the menu, choose an angle to rotate through.

3 Make a simple design in the sector which appears, and colour it.

4 Press finish to see your creation.

A A

AD D

DB B

BC C

C

centre of rotation

180° 360°

rotation rotation

DEMOThe rectangle has order of rotational symmetry of since it moves backto its original position under rotations of and

Click on the icon to find the order of rotational symmetry for anequilateral triangle.

2180 360o o:

ROTATING

FIGURES


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A translation of a figure occurs when every point on the figure is moved the same distance

in the same direction.

Under a translation the original figure and its image are congruent.

In the translation shown, the original figure has

been translated 4 units right and 3 units down

to give the image.

EXERCISE 22C

1 Describe each of the following translations:

a b c

d e

2 For the given figures, describe the

translation from:

a A to B b B to A

c B to C d C to B

e A to C f C to A

TRANSLATIONSC

DEMO

4

3

A'

object

image

A

A

B

C

object

image

object

image

object

image

objectimage

object

image


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3 Copy onto grid paper and translate using the given directions:

a 3 right, 4 down b 6 left, 4 up c 2 right, 5 up

We are all familiar with enlargements in the form of photographs or looking through a

microscope or telescope. Plans and maps are examples of reductions. The size of the image

has been reduced but the proportions are the same as the original. Most photocopiers can

perform enlargements and reductions.

The following design shows several enlargements:

In any enlargement or reduction, we multiply the lengths in the object by the scale factor

to get the lengths in the image.

Look at the figures in the grid below:

For the enlargement with scale factor 2, lengths have been doubled.

For the enlarement with scale factor 3, lengths have been trebled.

If shape B is reduced to shape A, the lengths are halved and the scale factor is 12

.

If shape D is reduced to shape A, the lengths are quartered and the scale factor is 14

.

A scale factor is: ² greater than 1 for an enlargement

² less than 1 for a reduction.

ENLARGEMENTS AND REDUCTIONSD

DEMO

DEMOA B C D

enlargement

scale factor 2enlargement

scale factor 3


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ACTIVITY 4 ENLARGEMENT BY GRIDS

You will need: Paper, pencil, ruler

What to do:

1 Copy the picture alongside.

2 Draw a grid 5 mm by 5 mm over the top

of the dog as shown alongside:

3 Draw a grid 10 mm by 10 mm alongside the grid

already drawn.

4 Copy the dog from the smaller grid onto the larger

grid. To do this accurately, start by transferring points

where the drawing crosses the grid lines. Then join

these points and finish the picture.

5 Use this method to change the size of

other pictures. You may like to try

making the picture smaller as well as

larger, by making your new grid smaller

than the original.

Enlarge using a scale factor of:

a 2 b 12

a b

EXERCISE 22D

1 In the following diagrams, A has been enlarged to B. Find the scale factor.

a b c



PRINTABLE

WORKSHEET

2

4

2

1

AB

A

BA

B

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2 In the following diagrams, A has been reduced to B. Find the scale factor.

a b c

3 Enlarge or reduce the following objects by the scale factor given:

a b c

d e f

4 Find the scale factor when A is transformed to B:

a b c

5 For each grid in 4, write down the scale factor which transforms B into A.

A tessellation is a pattern made using figures of the same shape and size. They must cover

an area without leaving any gaps.

The photograph alongside shows a tessellation of bricks

used to pave a footpath.

Tessellations are also found in carpets, wall tiles, floor

tiles, weaving and wall paper.

TESSELLATIONSE


B

ABA

A

B

PRINTABLE

WORKSHEET

scale factor 2 scale factor 3 scale factor Qw_

scale factor Qe_ scale factor 4 scale factor 2

A

B

AB

A B

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ACTIVITY 5 PAVING BRICKS

The following tile patterns are all tessellations:

What to do:

1 Using the “2£ 1” rectangle , form at least two different

tessellation patterns. One example is:

2 Repeat 1 using a “3£ 1” rectangle .

Draw tessellations of the following shapes.

a b

a b



For a tessellationthe shapes mustfit together with

no gaps.

This brick design is as it isconstructed from two different brick sizes.

not a tessellation

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cpurtle

Rectangle

ACTIVITY 6 CREATING TESSELLATIONS

DISCUSSION IN GOOD SHAPE

EXERCISE 22E

1 Draw tessellations using the following shapes:

a b c

2 Draw tessellations using the following shapes:

a b c

What to do:

Follow these steps to create your own tessellating pattern.

Step 1: Draw a square. Step 2: Cut a piece from one side and ‘glue’

it onto the opposite side.

Step 3: Rub out any unwanted

lines and add features.

Step 4: Photocopy this several times and cut

out each face. Combine them to

form a tessellation.

Make your own tessellation pattern and produce a full page pattern with 3 cm by 3 cm

tiles. Be creative and colourful. You could use a computer drawing package to do this

activity.

1 Research the shape of the cells in a beehive.

Explain why they are that shape.

2 Look at the shapes of paving blocks.

Explain what advantage some shapes have

over others. When building walls, what are

the advantages of rectangular bricks over

square bricks?


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ACTIVITY 7 COMPUTER TRANSFORMATIONS


What to do:

1 Pick a shape and learn how to translate, reflect and

rotate it.

2 Create a tessellation on your screen and colour it.

3 Print your final masterpiece.

² angle of rotation ² axis of rotation ² centre of rotation

² congruent ² enlargement ² image

² line of symmetry ² mirror line ² object

² reflection ² rotation ² rotational symmetry

² scale factor ² tessellation ² translation

TESSELLATIONS

BY COMPUTER

KEY WORDS USED IN THIS CHAPTER

ACTIVITY 8 DISTORTION TRANSFORMATIONS

In this activity we copy pictures onto unusual graph paper to produce

distortions of the original diagram. For example,

What to do:

1 On ordinary squared paper draw a picture of your own choosing.

2 Redraw your picture on different shaped graph paper. For example:

PRINTABLE

GRIDS

on becomes

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REVIEW SET 22A


5 a Translate the figure three units

left and one unit up.

b Enlarge the figure with

scale factor 2.

6 In the diagram A has been reduced to B.

Find the scale factor.

7 Draw tessellations of the following shapes.

a b

B

A

1 Draw the mirror image of:

a b

2 Draw the lines of symmetry for: 3 Rotate the given figure about O

through 90o anticlockwise.

4 For the given shape:

a draw the lines of symmetry

b state the order of rotational symmetry.

mirror line mirror line

O

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REVIEW SET 22B


4 a Find the order of rotational

symmetry for:

b Rotate the given figure 180o

about O.

5 Translate the given figure one unit

to the right and 3 units down.

6 Draw a tessellation using the given

shape.

O

1 Draw the mirror image of:

a b

2 Draw the lines of symmetry for:

a b

3 a Rotate the figure shown through

90o anticlockwise about O.

b Enlarge the figure with scale

factor 13

.

mirror line

O

mirror line

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transformations - mrs. purtle's math classes · reflections and line symmetry rotations and...

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