Topic 8 Transformation

INTRODUCTION

Symmetries create patterns that help us organise our world conceptually. Symmetric patterns occur in nature, and are invented by artists, craftspeople, musicians, choreographers and mathematicians. A pattern is ssymmetric if there is at least one line of symmetry (rrotation, translation, reflection, glide reflection) that leaves the pattern unchanged. Refer to Figure 8.1 for an example of symmetry in an equilateral hexagon pattern.

LEARNING OUTCOMES

By the end of this topic, you should be able to teach your studentshow to:

1. Identify a transformation as a one-to-one correspondencebetween points on a plane;

2. Determine the image of an object under a given transformation;

3. Determine the image of a point under a given transformation;and

4. Apply the concepts in coordinate geometry to determine theimage of an object.

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88 Transformation

TOPIC 8 TRANSFORMATION 190

Figure 8.1: Equilateral hexagon Pedagogical Content Knowledge Teachers are the key to changing the learning environment in the classroom. Therefore, you have to plan classroom experiences and create a supportive environment for learning to take place. Through classroom practices, you have to promote students' mmathematical reasoning, challenge them with problems through which they learn to value mathematics, and provide them with a strong foundation for further study. This topic will help you to teach students to investigate basic geometric relationships, ccongruence and ssimilarity, and identify ggeometric transformations. They should know that a figure, picture or pattern is said to be ssymmetric if there is at least one line of symmetry that leaves the figure unchanged. For example, the letters in AAMOYOMA form a symmetric pattern. If your students draw a vertical line through the centre of "Y" and then reflect the entire phrase across the line, the left side becomes the right side, and vice versa. The picture does not change.

TERMS AND CONCEPTS (OBJECT AND IMAGE)

In order to teach this topic effectively, you need to consider the importance of oral and mental work in all parts of the mathematics lesson. You can explain the basic concepts of transformation by using the basic types of symmetry. (a) RRotation by 90 degrees about a fixed point is an example of pplane

symmetry.

8.1


(b) Another basic type of symmetry is rreflection. The reflection of a figure on the plane about which a line moves its reflected image can be seen if you view it using a mirror placed on the line. Another way to make a reflection is to fold a piece of paper and trace the figure onto the other side of the fold.

(c) A third type of symmetry is ttranslation. Translating an object means moving it without rotating or reflecting it. You can describe a translation by stating how far it moves an object, and in what direction.

(d) The fourth (and last) type of symmetry is gglide reflection. A glide reflection combines a reflection with a translation along the direction of the mirror line.

You can further explain to your students about object and image by using this example: Identify the symmetric relationship in two-dimensional objects.

Example 1:

8.1.1 The Basic Properties of Equilaterals

You may begin a class by explaining the basic properties of equilaterals using a set of regular polygons. Regular Polygons Example 2:

(a) A regular polygon with three sides is called an equilateral triangle. Each angle is 60 degrees (Figure 8.2(a)).

(b) A regular polygon with four sides is called a square. Each angle is 90 degrees (Figure 8.2(b)).

(c) A regular polygon with five sides is called a pentagon. Each angle is 108 degrees (Figure 8.2(c)).

(d) A regular polygon with six sides is called a hexagon. Each angle is 120 degrees (Figure 8.2(d)).


Figure 8.2: Regular polygons Equilateral Triangles You may read the following instructions and ask your students to imagine the final image first before sketching it.

Instruction to Students Image

Imagine an equilateral triangle. Imagine another identical equilateral triangle. Place it alongside the original so that the edges match exactly.

What is the name of the shape you have made?

Is there more than one possibility?

Explain.

Take another identical equilateral triangle and add this to the figure.

What is the name of the shape you have made?

Describe its properties as accurately as you can.

Is there more than one possibility?

Explain.

Take four identical equilateral triangles.

How many different shapes can you make by joining the edges?

Describe as accurately as possible one of the shapes you have made from the four equilateral triangles.


8.1.2 Concepts of Isometry and Congruence

To introduce isometry/similarity and congruence to students, you may carry out the following activity. Opening Activity: Ask your students:

(a) What are ccongruent shapes? These are shapes that have the same size and shape.

(b) What are ssimilar shapes? These are shapes that have the same shape but not the same size.

Let students review and practise how to differentiate congruent and similar shapes by using the ttangram pieces. Ask students to indicate and record the pieces that are congruent (triangles that are the same size and shape), and then the pieces that are similar (triangles that are of various sizes). Let students work together in groups to create congruent and similar shapes by combining the tangram pieces. Next, explain further to your students the definition of isometric/similarity and congruence. Definition 1: Informally, two geometric objects on the plane are ssimilar if they have the same shape. More formally, having the same shape means that one figure can be mapped onto the other.

Replace the equilateral triangles by squares or by right-angled triangles. Ask your students the following questions:

1. What happens if you do the same transformation twice?

2. How many combinations of two transformations are there?

3. What happens if you combine more than two transformations?

SELF-CHECK 8.1


Help students to identify and apply conditions that are sufficient to guarantee the ssimilarity of triangles: (a) Identify two triangles as similar if the ratios of the lengths of corresponding

sides are equal (sside-side-side or SSSS criterion/postulate), if the ratios of the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (sside-angle-side or SSAS criterion), or if two pairs of corresponding angles are congruent (aangle-angle or AAA criterion).

(b) Apply the SSS, SAS and AA criteria to verify whether or not two triangles are similar.

(c) Apply the SSS, SAS and AA criteria to construct a triangle similar to a given triangle using a straight edge and a compass or geometric software.

(d) Identify the constant of proportionality and determine the measures of corresponding sides and angles for similar triangles.

(e) Demonstrate that the rate of change (slope) associated with any two points on a line is a constant by using similar triangles.

(f) Recognise that a line drawn inside a triangle parallel to one side forms a smaller triangle similar to the original one; explain why this is true and apply this knowledge.

Definition 2: Informally, two figures on the plane are ccongruent if they have the same size and shape. More formally, having the same size and shape means that one figure can be mapped into the other. When formally defining the concepts of iisometry and ssimilarity transformation, the students need to apply their knowledge of geometric transformations. In the process, they will discover the minimum conditions required for triangles to be isometric or similar (students can discover this through experimentation or observation), as well as the properties of isometric or similar plane figures. Here, the students can once again apply their knowledge of ratios and proportions. EEquivalent plane figures will be defined as "ffigures with the same area" and their main properties will be studied.


The following examples will help students to develop their ability in analysing geometric problems. Example 3: Is ABC EFD an isometry?

Solution: From the figure, we can observe that the corresponding image vertices of the pre-image vertices A, B and C are E, F and D, respectively.

In triangles ABC and EDF, AC = ED = 5 BC = FD = 13

ACB EDF Thus, by the SSAS postulate, the two triangles are congruent, and the transformation is an isometry. Example 4: Consider the given figures. If it is possible to prove that the triangles are congruent, write the congruence and identify whether the SAS postulate or the SSS postulate is used. If not, write not possible.


Solution: Write congruence statements for ABC and DEF. CA = FD and CB = FE We have sufficient information to prove the congruence of the two triangles. Write the congruence.

ABC DEF Name the postulate that can be used to prove the triangles congruent. The three sides of one triangle are congruent to the three sides of the other triangle. So, use the SSS postulate. Explain to your students why ccongruence is a special case of similarity; determine and apply conditions that guarantee the congruence of triangles:

(a) Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (ttranslation, rreflection or rrotation).

(b) Identify two triangles as congruent if two pairs of corresponding angles and their included sides are all equal (aangle-side-angle or AASA criterion).

After your students have completed the above activity, discuss the concepts they have just learned. Assess their understanding and clarify their doubts before moving on to the next section.

8.1.3 Coordinates and Transformation

Let us continue with the study of transformation so that students can become proficient at visualising and recognising transformed figures on the coordinate plane.

Ask students to briefly provide the definitions of isometry and congruence.

SELF-CHECK 8.2


A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face the same direction. (a) A translation is a movement up, down, left or right without changing size,

flipping or turning. We can specify the translation that has occurred by stating how many units the object has moved vertically (up or down) and horizontally (left or right).

(b) When finding the total translations, movements (the number of units) to the right and up are considered positive, while movements to the left and down are treated as negative.

Let us examine some translations related to ccoordinate geometry. Example 5: Examine how each vertex moves the same distance and in the same direction.

Example 6: In this example, the "slide" moves the figure 7 units to the left and 3 units down.


Description: Mapping:

Symbol:

7 units to the left and 3 units down. (x, y) (-7, -3) (This is read: "the x and y coordinates will become x 7 and y 3". Notice that mmovements to the left and down are negative, while mmovements to the right and up are positive just as it is on coordinate axes.)

7, 3T

(The -7 tells you to subtract 7 from all of your x-coordinates, while the -3 tells you to subtract 3 from all of your y-coordinates.)

Example 7: Use the graph paper to graph the quadrilateral: S(-5, -2), T(-4, 1), O(-1, -2) and P(-3, -5). Write an explanation of what happens to any object if you add 4 to the first coordinate of every point and 2 to every second coordinate. What will be the new coordinates for S1, T1, O1 and P1? Solution:

Pre-Image/Object IImage

x y x y

S -5 -2 S1 -1 -4

T -4 1 T1 0 -1

O -1 -2 O1 3 -4

P -3 -5 P1 1 -7

If we add 4 to every first coordinate, the pre-image will move 4 units to the right. If we add -2 to every second coordinate, the pre-image will move down 2 units. Ask your students to plot the graph (object and image). Example 8: The coordinates of a translated triangle are F '(-8, -16), T '(-7, -13) and H '(-1, -14). If the coordinates of F are (-3, -12), what are the coordinates of T and H?


Solution: From points F and F ', we know that -3 = -8 + 5 and -12 = -16 + 4. Let T = 1 1,x y and H = 2 2,x y . We obtain:

1 -7 5 -2x and 1 -13 4 -9y , thus T = (-2, -9). 2 -1 5 4x and 2 -14 4 -10y , thus H = (4, -10).

BASIC CONCEPTS OF TRANSFORMATION

Let the students know what it is they will be learning today. Say something like this: Today, class, we will learn what translations, reflections and rotations are to a mathematician. You may then use a diagram or picture to assess how well the students understand the different transformations they have learned and how well they can use the associated mathematical terms. You may also suggest that they practise their skills by using the Innteractive Transmographer. (Refer to http://www.shodor.org/interactivate/activities/Transmographer).

8.2.1 Reflection

The four Euclidean isometries reflection, translation, rotation and glide reflection can be expressed as compositions of reflections. In the following, students will find some texts, pictures and animations illustrating the relationship between reflection and other isometries.

8.2

1. What would the coordinates of point A' be if triangle ABC located at A(1,1), B(4,1) andC(2,3) was translated left 3 units?

2. What is the image translation of triangle KDH with coordinates K(9, 8), D(3, 1) and H(0, 11) that are translated (2,-1)?

3. The coordinates of a translated triangle are T ' (-5, -3), D' (2, 9) and M' (0, -10). If the coordinates of T are (-1,-1), what are the coordinates of D and M?

ACTIVITY 8.1


Example 9: Do the following diagrams illustrate line reflections? (a)

(b)

(c)

(d)


Explain to your students that to reflect an object means to produce its mirror image. Every reflection has aa mirror line. A reflection of an "R" is a backwards "R". Example 10:

8.2.2 Rotation (0 360)

To rotate an object means to turn it around. Every rotation has a ccentre and aan angle. Example 11:

Example 12:


8.2.3 Translation

To translate an object means to move it without rotating or reflecting it. Every translation has a ddirection and a ddistance. Example 13: Do the following diagrams illustrate translations?

Example 14:

Example 15:


8.2.4 Glide Reflection

A glide reflection ccombines a reflection with a translation along the direction of a mirror line. Glide reflection is the only type of symmetry that involves more than one step. Example 16:

Example 17:

Have your students practise answering the questions in the following activity and ask them to give reasons for their answers.


1. This drawing has been rotated 180o. Answer Yes or No.

2. This triangle has been rotated in a clockwise direction. True or

False?

3. Seahorse 2 is a 90-degree counterclockwise rotation of Seahorse

1. True or False?

4. Why does the word AMBULANCE appear in the following form

on the front of rescue vehicles?

ACTIVITY 8.2


THE PROPERTIES OF TRANSFORMATION

Once your students have mastered the concepts well, you can gradually introduce some examples of transformation.

In this subtopic, you will learn how to help your students to rrecognise and visualise transformations and lines of symmetry of two-dimensional shapes.

8.3.1 Translation

Definition 3: A translation is a transformation that slides each point of a figure the same distance and direction. This is the result when a figure is reflected over a pair of parallel lines. Translation has a multitude of applications in our everyday life. In the field of mathematics, translation can help us to understand the transformation of algebraic functions. Translation plays an important part in ggraphic design and mmanufacturing. We are surrounded by samples of translation in the design of fabric, wallpaper and floor tiling. Examples of translation can also be found in sheet music. Figure 8.3 illustrates how the pre-image (triangle ABC) is reflected over a pair of parallel lines (m and n). The result of the two reflections is the image of the translation (triangle A''B ''C '').Notice that the distance between A and A'' is twice the distance between the parallel lines, and the line AA'' is perpendicular to lines m and n. Example 18: Properties of translation (Figure 8.3): A translation preserves congruence. A translation preserves orientation.

8.3


Figure 8.3: Properties of translation

Notice that the pre-image (triangle ABC) and the image (triangle ABC) are congruent (all corresponding sides and angles are congruent). Also, if you look at the vertices of triangle ABC in alphabetical order, they are arranged in aa clockwise orientation. The vertices of triangle ABC are also arranged in a clockwise orientation.

8.3.2 Rotation

Point out to your students that they have to identify the properties of rotation. Encourage them to form class discussion.

Definition 4: A rrotation is a transformation in which every point of a figure moves along a circular path around a fixed point that is called the ccentre of rotation. A figure can be rotated about a point called the centre of rotation. To specify the rotation, we need to give the angle through which the objects are to be turned, and the direction of the rotation (clockwise or anticlockwise).

Lines that are drawn from a point and its image to the centre of rotation form an angle that is always the same measure. This angle is called the aangle of rotation. To


ensure a better understanding of this concept of rotation, ask your students to experiment with rotating figures on the coordinate plane.

They can observe rrotational symmetry on the face of a clock, a windmill and the tyres of cars. Designs created by rotation can be found in quilts, fabrics, rugs and various logos. In the field of geometry, regular polygons have rotational symmetry.

The same result can be achieved by reflecting a figure over two intersecting lines (Figure 8.4). Polygon ABCDE is first reflected over line m before it is reflected over line n. The image of the rotation is ABCDE. The angle of rotation can be found by measuring the angle formed by the segments that connect points A and A'' to point J (the centre of rotation).

Example 19: Properties of rotation (Figure 8.4): A rotation is a transformation that preserves congruence. A rotation preserves orientation.

Figure 8.4: Properties of rotation

Notice that polygon ABCDE is congruent to polygon ABCDE (the corresponding parts of each polygon are congruent). Also, the vertices of both polygons ABCDE and ABCDE are in a counterclockwise orientation.


Class Activity: RRotational Art The students will create art by rotating an object(s) several times around a fixed point of rotation. The art will be defined by colouring it in. The students will exchange pictures and try to determine the centre, angle and direction of the rotations.

8.3.3 Reflection

Explain the following definition and examples to your students. Definition 5: A rreflection is a transformation in which each point of a figure has an image that is the same distance from the line of reflection as the original figure. Reflections are often called mmirror images and the lines in which the objects are reflected are called mmirror lines. Mirror images always have reverse orientation. The points of the image and the corresponding points of the object lie on the same line, which is perpendicular to the mirror line. Each point of the image is the same distance from the mirror line as each point of the object on the other side of the mirror.

The concept of reflection surrounds us in our everyday life. We can find reflection symmetry in architecture, nature, sports, graphic design and everyday objects. We can see our own reflection by using a mirror or looking into a pool of water. Kaleidoscopes and periscopes use reflections to produce beautiful symmetric designs.

In Figure 8.5, every point in triangle ABC is mapped onto its image in triangle A'B 'C '. The reflection line m is the pperpendicular bisector of the segments that connect each point from the original triangle with each point in the image (m is the perpendicular bisector of segments AA', BB ' and CC ').

Example 20: Properties of reflection (Figure 8.5):

A reflection is a transformation that preserves congruence.

A reflection changes orientation.

The line of reflection is the perpendicular bisector of every segment that connects a point and its image.


Figure 8.5: Properties of reflection

Notice that the triangles are congruent (all corresponding sides are congruent and all corresponding angles are congruent). If we look at the vertices of triangle ABC in alphabetical order, they are arranged in a counterclockwise orientation. The vertices of triangle A'B'C' are arranged in a clockwise orientation. Students can observe these basic properties of reflection when they complete the reflection investigation using the GGeometer's Sketchpad. They can also verify their results using simple ppaper folding activities. Ask students to think of other ways to demonstrate the properties of reflection. Class Activity: MMiming Reflection Each student will be paired with a partner. The partners will face each other and be told to reflect or mimic what their partner is doing. You need to start slow and then move on to doing it faster. When finished, you may discuss the properties of reflection, specifically pointing out that whatever the students did with their left hands, their partners reflected it with their right hands, and vice versa. The following are some questions for in-class activity. Encourage your students to form class discussion.


1. Which of the following lettered figures are translations of the shape of the ffirst arrow given?

2. Can the following diagram be used to illustrate a translation of ABC three units to the right and two units up?

ACTIVITY 8.3


3. Which of the following translations best describes the diagram below?

(a) 3 units right and 2 units down; or

(b) 3 units left and 2 units up.

4. Which statement best describes the translation shown in the

diagram below?

(a) A horizontal translation;

(b) A vertical translation; or

(c) A translation of 2 units right and 1 unit down.

5. A person is sitting 4m away from a mirrored wall. How far away

from the person will his or her reflection appear to be? 6. Polygon A'B'C'D' is a 180 counter-clockwise rotation of polygon

ABCD. True or False?


7. Fish 2 is a 45 counter-clockwise rotation of Fish 1. True or False?

8. (a) What word will be seen when MOM is reflected over line m?

(b) What word will be seen when MOM is reflected over line l?

9. Which of the following shows a reflection, translation and rotation?


Discuss with your colleague other activities related to daily life which can help your students to apply their mathematical knowledge and communicate their mathematical thinking coherently and clearly.

A ttransformation that preserves distance is an iisometry. A ddirect isometry also preserves orientation or order.

An iindirect or opposite isometry changes the order (such as from clockwise to counterclockwise lettering).

A line rreflection creates a figure that is congruent to the original figure and is an isometry. Since naming the figure in a reflection requires changing the order, it is an indirect or opposite isometry.

A translation creates a figure that is congruent to the original figure and preserves distance and orientation it is a direct isometry.

A rotation creates a figure that is congruent to the original figure and preserves distance and orientation it is a direct isometry.

AA criterion

Angle

Centre

Clockwise orientation

Congruence

Congruent

Coordinate geometry

Counterclockwise orientation

Direct isometry

Equilateral

Geometer's Sketchpad

Opposite isometry

Orientation

Perpendicular bisector

Plane symmetry

Postulate

Pre-image

Preserves distance

Reflection

Rotation

SAS criterion

Similarity


Geometric transformations

Glide reflection

Graphic design

Image

Isometric

Isometry

Mirror line

Object Distance formula

Graphical method

Horizontal axis

Hypotenuse

Midpoint

Midpoint formula

SSS criterion

Symmetric

Symmetry

Tangram pieces

Transformation

Translation

Transmographer

Two-dimensional shapes

Cheong, Q. L, & Teh, W. L. (2008).Essential Mathematics Form 2. Petaling Jaya: Pearson Malaysia.

Hearn, D., & Baker, P. M. (1997).Computer graphics (2nd ed.). New Delhi, India:

Prentice Hall. Lee, L. M. (2007). Mathematics Form 2. Shah Alam: Arah Pendidikan. Walter, M. (2006). Geometry and its applications. New York, NY: Academic

Press.

Topic 8 Transformation

Documents

example of symmetry

line of symmetry rrotation

transformation topic

basic type of symmetry

basic types of symmetry

example of pplane symmetry

mirror line

concepts object