Transformation in Geometry

Transformation A transformation changes the position or size of a shape on a coordinate plane

TRANSFORMATIONS CHANGE THE POSTION OF A SHAPE CHANGE THE SIZE OF A SHAPE TRANSLATION ROTATION REFLECTION DILATION Change in location Turn around a point Flip over a line Change size of a shape

Renaming Transformations It is common practice to name transformed polygon using the same letters with a “prime” symbol: It is common practice to name a polygon using capital letters for each of its vertices: The original polygon is often called the Pre-image The transformed polygon is often called the image

Translation A transformation that moves each point in a figure the same distance in the same direction.

In a translation a figure slides up or down, or left or right. No change in shape, size or the direction it is facing. The location is the only thing that changes. They are sometimes called “slides” In graphing translation, all x and y coordinates of a translated figure change by adding or subtracting.

Translation When an object is moved in a straight line in a given direction we say that it has been translated. For example, we can translate triangle ABC 5 squares to the right and 2 squares up: C’ A’ B’ image C A B object C A B object C A B object C A B object C A B object C A B object C A B object C A B object object Introduce the key terms and ask pupils to give examples of translations from everyday situations. For example, walking in a straight line from one side of the room to the other. Discuss how to define a translation. There are two ways to mathematically describe movement in a straight line in a given direction. One is to to give the direction as an angle and a distance (see S7 Measures – bearings), the other is to give the direction using a square grid. A movement in a straight line can be given as a number or units to the left or right and a number of units up or down. Stress that when a shape is translated each vertex (or corner) is moved through the given translation. The movement across is always given before the movement up or down (compare with coordinates). Ask pupils how we could return the translated shape back to its original position. Every point in the shape moves the same distance in the same direction.

Describing translations When we describe a translation we always give the movement left or right first followed by the movement up or down. We can describe translations using function notation. For example, describes a translation of triangle ABC as 3 right and 4 down. As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left. Compare a translation given as an angle and a distance to using bearings. For example, an object could be translated through 5cm on a bearing of 125º.

Translations on a coordinate grid y A(5, 7) The vertices of a triangle lie on the points A(5, 7), B(3, 2) and C(–2, 6). C(–2, 6) 7 6 5 4 3 2 B(3, 2) Translate the shape 3 squares left and 8 squares down. Label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 A’(2, –1) –2 C’(–5, –2) What do you notice about each point and its image? –3 Point out that when we translate shapes on a coordinate grid we can find the coordinates of the vertices of the image shape by adding to the original’s x-coordinates the size of the translation to the right (or subtracting the size of a translation to the left); and adding to the original’s y-coordinates the size of the translation up (or subtracting the size of a translation down). –4 –5 –6 B’(0, –6) –7

Translations on a coordinate grid y The coordinates of vertex A of this shape are (–4, –2). 7 6 5 4 3 When the shape is translated the coordinates of vertex A’ are (3, 2). 2 A’(3, 2) 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 –2 A(–4, –2) What translation will map the shape onto its image? –3 Tell pupils that we can work this out by subtracting the coordinate of the original shape from the coordinate of the image. The difference between the x-coordinates is 7 (3 – –4) and the difference between the y-coordinates is 4 (2 – –2). The required translation is therefore 7 right and 4 up. –4 –5 –6 –7 7 right 4 up

Translations on a coordinate grid 1 2 3 4 5 6 –2 –3 –4 –5 –6 –7 –1 y x 7 The coordinates of vertex A of this shape are (3, –4). When the shape is translated the coordinates of vertex A’ are(–3, 3). A’(–3, 3) What translation will map the shape onto its image? The difference between the x-coordinates is –6 (–3 – 3) and the difference between the y-coordinates is 7 (3 – –4). The required translation is therefore 6 left and 7 up. If required this slide can be copied and modified to produce more examples. To change the shapes, go to View/toolbars/drawing to open the drawing toolbar. Select the shape and using the drawing toolbar, click on draw/edit points to drag the points to different positions. Edit the new coordinates as necessary. A(3, –4) 6 left 7 up

Translation golf In this game pupils are required to estimate the required translation to move the ball to the hole. The length of one unit is shown in the top right-hand corner. To translate the ball to the left or down, a negative number must be used. Challenge pupils to get the ball into the hole in as few moves as possible.

Reflection A transformation where a figure is flipped across a line such as the x-axis or the y-axis.

In a reflection, a mirror image of the figure is formed across a line called a line of symmetry. No change in size. The orientation of the shape changes. In graphing, a reflection across the x -axis changes the sign of the y coordinate. A reflection across the y-axis changes the sign of the x-coordinate.

Reflection An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example, Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.

The image is congruent to the original shape. Reflecting shapes If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A B C D A’ B’ object image C’ Explain that we call the original shape the object and the reflected shape the image. The image of an object can be produced by any transformation including rotations, translations and enlargements, as well as reflections. Define the word congruent to mean the same shape and size. Link: S2 2-D shapes – congruence. D’ mirror line or axis of reflection The image is congruent to the original shape.

Reflecting shapes If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line. A B C D A’ B’ C’ D’ object image mirror line or axis of reflection Stress that the mirror line is always perpendicular (at right angles) to any line connecting a point to its image. The mirror line also bisects the line (divides it into two equal parts).

Reflecting shapes Use this activity to dynamically prove the result on the previous slide, regardless of the shape or the position of the mirror line.

Reflecting shapes using tracing paper Suppose we want to reflect this shape in the given mirror line. Use a piece of tracing paper to carefully trace over the shape and the mirror line with a soft pencil. When you turn the tracing paper over you will see the following: Place the tracing paper over the original image making sure the symmetry lines coincide. Draw around the outline on the back of the tracing paper to trace the image onto the original piece of paper.

Reflect this shape Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon. If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

reflectional symmetry. Sometimes, a figure has reflectional symmetry. This means that it can be folded along a line of reflection within itself so that the two halves of the figure match exactly, point by point. Basically, if you can fold a shape in half and it matches up exactly, it has reflectional symmetry.

REFLECTIONAL SYMMETRY Line of Symmetry Reflectional Symmetry means that a shape can be folded along a line of reflection so the two haves of the figure match exactly, point by point. The line of reflection in a figure with reflectional symmetry is called a line of symmetry. The two halves are exactly the same… They are symmetrical. The two halves make a whole heart.

REFLECTIONAL SYMMETRY The line created by the fold is the line of symmetry. How can I fold this shape so that it matches exactly? A shape can have more than one line of symmetry. Where is the line of symmetry for this shape? I CAN THIS WAY NOT THIS WAY Line of Symmetry

REFLECTIONAL SYMMETRY How many lines of symmetry does each shape have? 3 4 5 Do you see a pattern?

REFLECTIONAL SYMMETRY Which of these flags have reflectional symmetry? United States of America Canada No England No Mexico

Rotation A transformation where a figure turns about a fixed point without changing its size and shape.

In a rotation, figure turns around a fixed point, such as the origin. No change in shape, but the orientation and location change. Rules for 90 degrees rotation about the origin- Switch the coordination of each point. Then change the sign of the y coordinate. Ex. A (2,1) to A’ ( 1,-2)

Describing a rotation A rotation occurs when an object is turned around a fixed point. To describe a rotation we need to know three things: The angle of rotation. For example, ½ turn = 180° ¼ turn = 90° ¾ turn = 270° The direction of rotation. For example, clockwise or counterclockwise. The center of rotation. This is the fixed point about which an object moves.

What does a rotation look like? center of rotation A ROTATION MEANS TO TURN A FIGURE

This is another way rotation looks The triangle was rotated around the point. This is another way rotation looks center of rotation A ROTATION MEANS TO TURN A FIGURE

If a shape spins 360, how far does it spin? ROTATION If a shape spins 360, how far does it spin? 360 All the way around This is called one full turn.

Half of the way around ROTATION If a shape spins 180, how far does it spin? Rotating a shape 180 turns a shape upside down. Half of the way around 180 This is called a ½ turn.

If a shape spins 90, how far does it spin? ROTATION If a shape spins 90, how far does it spin? One-quarter of the way around 90 This is called a ¼ turn.

ROTATION Describe how the triangle A was transformed to make triangle B A B Triangle A was rotated right 90 Describe the translation.

ROTATION Describe how the arrow A was transformed to make arrow B B A Arrow A was rotated right 180 Describe the translation.

to spin a shape the exact same ROTATION When some shapes are rotated they create a special situation called rotational symmetry. to spin a shape the exact same

ROTATIONAL SYMMETRY A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape. Here is an example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? 90 Yes, when it is rotated 90 it is the same as it was in the beginning. So this shape is said to have rotational symmetry.

So this shape is said to have rotational symmetry. A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape. Here is another example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? Yes, when it is rotated 180 it is the same as it was in the beginning. So this shape is said to have rotational symmetry. 180

Here is another example… ROTATIONAL SYMMETRY A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape. Here is another example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? No, when it is rotated 360 it is never the same. So this shape does NOT have rotational symmetry.

Does this shape have rotational symmetry? ROTATION SYMMETRY Does this shape have rotational symmetry? Yes, when the shape is rotated 120 it is the same. Since 120  is less than 360, this shape HAS rotational symmetry 120

Rotational symmetry An object has rotational symmetry if it fits exactly onto itself when it is turned about a point at its centre. The order of rotational symmetry is the number of times the object fits onto itself during a 360° turn. If the order of rotational symmetry is one, then the object has to be rotated through 360° before it fits onto itself again. Only objects that have rotational symmetry of two or more are said to have rotational symmetry. Discuss what it means for a shape to have rotational symmetry.

Finding the order of rotational symmetry Use this activity to show how we can find the order of rotational symmetry using tracing paper. Explain that if we draw an arrow to indicate the top of the paper before we start, then we can tell when we’ve completed a full turn. For each shape, turn the paper through 360º and count how many times the shape on the tracing paper coincides with the shape underneath. This number is the order of rotational symmetry. Warn pupils not to count the first position twice. If the order of rotational symmetry is 1, the shape does not have rotational symmetry. Ask pupils if any of the shapes have line symmetry.

Rotational symmetry What is the order of rotational symmetry for the following designs? Decide on the order of rotational symmetry before revealing the answer. Rotational symmetry order 4 Rotational symmetry order 3 Rotational symmetry order 5

Determining the direction of a rotation Sometimes the direction of the rotation is not given. If this is the case then we use the following rules: A positive rotation is an counterclockwise rotation. A negative rotation is an clockwise rotation. For example, A rotation of 60° = an anticlockwise rotation of 60° This is probably the opposite to what most people would expect. Explain that it is a convention that has been agreed between mathematicians all over the world. It’s just that, mathematically speaking, the hands of a clock turn in a negative direction! Establish that equivalent rotation can always be found by changing the sign and subtracting the angle from 360º. Ask pupils to give examples of other equivalent rotations. A rotation of –90° = an clockwise rotation of 90° Explain why a rotation of 120° is equivalent to a rotation of –240°.

Dilation A transformation where a figure changes size.

Dilation In dilation, a figure is enlarged or reduced proportionally. No change in shape, but unlike other transformation, the size changes. In graphing, for dilation, all coordinates are divided or multiplied by the same number to find the coordinates of the image.

Dilation Dilation is altering the size of the figure by k = n . The dilation of this figure is k = 3. The image is 3 times bigger than the preimage. DILATION – A transformation that alters the size of a figure, but not its shape.

Enlargement Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. Stress the every side in the shape has to be two times bigger to enlarge the shape by a scale factor of 2. Ask pupils to tell you the difference between the angles in the first shape (the object) and the angles in the second (the image). Tell pupils that when we enlarge a shape the lengths change but the angles do not. The original shape and its image are not congruent but they are similar (they are different sizes but are the same shape). We say that shape A has been enlarged by scale factor 2.

Enlargement When a shape is enlarged the ratios of any of the lengths in the image to the corresponding lengths in the original shape (the object) are equal to the scale factor. A’ A 6 cm 4 cm 6 cm 9 cm B B’ C 8 cm 12 cm C’ Remind pupils that we can write ratios as fractions as well as using the ratio notation. The notation that we use depends on the context of the problem. AC : A’C’ is the same ratio as AC/A’C’, written in a different way. The result means that the ratio of any two corresponding lengths in the object and the image can be used to find the scale factor. Click through to show how to find the scale factor using the lengths in the diagram. A’B’ AB B’C’ BC A’C’ AC = = = the scale factor 6 4 12 8 9 6 = = = 1.5

Congruence and similarity Is the image of an object that has been enlarged congruent to the object? Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal. In an enlarged shape the corresponding angles are the same but the lengths are different. The image of an object that has been enlarged is not congruent to the object, but it is similar. Review the meaning of the term congruent and introduce the term similar. Two shapes are called similar if they are the same shape but a different size. Link: S2 2-D shapes – congruence. In maths, two shapes are called similar if their corresponding angles are equal. Corresponding sides are different lengths, but the ratio in lengths is the same for all the sides.

Find the scale factor What is the scale factor for the following enlargements? B’ B Count squares to deduce that the scale factor for the enlargement is 3. Show that that the ratios of any of the corresponding lengths on the image and in the object are equal to the scale factor. Scale factor 3

Find the scale factor What is the scale factor for the following enlargements? C’ C By counting squares deduce that the scale factor for the enlargement is 2. Scale factor 2

Find the scale factor What is the scale factor for the following enlargements? D’ Deduce that the scale factor for the enlargement is 3.5 by counting squares. Point out that it doesn’t matter which lengths we compare. They are all enlarged by the same scale factor. For example, in the second shape it is difficult to determine the length of the sides. Instead we can compare the diagonals of the shapes. The size of the diagonal of the first shape is 2 units and the size of the diagonal of the second shape is 7 units. 7 ÷ 2 gives us the scale factor 3.5. D Scale factor 3.5

Find the scale factor What is the scale factor for the following enlargements? E E’ The lengths in the second shape are half the size of the lengths in the first shape and so the scale factor is 0.5 or ½. Point out that technically this is still called an enlargement even though the shape has been made smaller. Scale factor 1 doesn’t change the size; factors greater than 1 increase the size; and factors less than 1 make the shape smaller. Scale factor 0.5

Using a center of enlargement To define an enlargement we must be given a scale factor and a center of enlargement. For example, enlarge triangle ABC by scale factor 2 from the centre of enlargement O: A’ O A C B B’ The scale factor tells us the size of the enlargement and the centre of enlargement tells us the position of the enlargement. Here the scale factor is 2, so: The distance from O to A’ is twice the distance from O to A. The distance from O to B’ is twice the distance from O to B. The distance from O to C’ is twice the distance from O to C. C’ OA’ OA = OB’ OB = OC’ OC = 2

Using a centre of enlargement Enlarge parallelogram ABCD by a scale factor of 3 from the centre of enlargement O. A’ D’ A D O B C The distance from O to A’ is three times the distance from O to A. The distance from O to B’ is three times the distance from O to B. The distance from O to C’ is three times the distance from O to C. The distance from O to D’ is three times the distance from O to D. B’ C’ OA’ OA OB’ OB OC’ OC OD’ OE = = = = 3

Exploring enlargement Use this activity to dynamically explore the relationship between the position of the centre of enlargement and the position of the image. Demonstrate examples where the centre of enlargement is inside the shape, on an edge or on a vertex. Establish that the further the centre of enlargement is from the object the further the image is from the object. Reveal the lengths of some of the sides and ask pupils to find the missing lengths.

Enlargement on a coordinate grid y The vertices of a triangle lie on the points A(2, 4), B(3, 1) and C(4, 3). 10 9 A’(4, 8) 8 7 C’(8, 6) The triangle is enlarged by a scale factor of 2 with a centre of enlargement at the origin (0, 0). 6 5 A(2, 4) 4 C(4, 3) 3 2 When the centre of enlargement is at the origin, the coordinates of each point in the image can be found by multiplying the coordinates of each point on the original shape by the scale factor. B’(6, 2) 1 B(3, 1) What do you notice about each point and its image? 1 2 3 4 5 6 7 8 9 10 x

Coordinate Dilation Each point is multiplied by k to find the dilation.

Enlargement on a coordinate grid 1 2 3 4 5 6 7 8 9 10 y x The vertices of a triangle lie on the points A(2, 3), B(2, 1) and C(3, 3). A(6, 9) C’(9, 9) The triangle is enlarged by a scale factor of 3 with a centre of enlargement at the origin (0, 0). A(2, 3) C(3, 3) B’(6, 3) Using the rule from the previous slide, ask pupils to predict the coordinates of the image before revealing them. B(2, 1) What do you notice about each point and its image?

Combining reflections An object may be reflected many times. In a kaleidoscope mirrors are placed at 60° angles. Shapes in one section are reflected in the mirrors to make a pattern. How many lines of symmetry does the resulting pattern have? Ask pupils to tell you the order of the rotational symmetry. Ask pupils to tell you if mirrors placed like this would always produce a pattern with three lines of symmetry and rotational symmetry of order 3. What would happen if mirrors were placed at 45º angles? Ask pupils to design a kaleidoscope pattern with either 60º angles between the mirror lines (using isometric paper) or 45º angles between the mirror lines (using squared paper). Does the pattern have rotational symmetry?

Parallel mirror lines What happens when an object is reflected in parallel mirror lines placed at equal distances? Ask pupils to describe what has happened in terms of other transformations that they know. Establish that the object is translated as a result of being reflected and then reflected again.

Parallel mirror lines Suppose we have two parallel mirror lines M1 and M2. We can reflect shape A in mirror line M1 to produce the image A’. A A’ A’’ We can then reflect shape A’ in mirror line M2 to produce the image A’’. How can we map A onto A’’ in a single transformation? M1 M2 Reflecting an object in two parallel mirror lines is equivalent to a single translation.

Perpendicular mirror lines Suppose we have two perpendicular mirror lines M1 and M2. We can reflect shape A in mirror line M1 to produce the image A’. A’ A M2 We can then reflect shape A’ in mirror line M2 to produce the image A’’. A’’ How can we map A onto A’’ in a single transformation? M1 Reflection in two perpendicular lines is equivalent to a single rotation of 180°.

Combining rotations Suppose shape A is rotated through 100° clockwise about point O to produce the image A’. A Suppose we then rotate shape A’ through 170° clockwise about the point O to produce the image A’’. 100° A’’ O A’ 170° How can we map A onto A’’ in a single transformation? To map A onto A’’ we can either rotate it 270° clockwise. Two rotations about the same centre are equivalent to a single rotation about the same centre.

Combining translations Suppose shape A is translated 4 units left and 3 units up. Suppose we then translate A’ 1 unit to the left and 5 units down to give A’’. A’ How can we map A to A’’ in a single transformation? A A’’ We can map A onto A’’ by translating it 5 units left and 2 units down. Point out to pupils that we can quickly translate a shape by considering just one of its points (in this case, the vertex shown by an orange cross). Discuss how a translation 4 units left and 3 units up followed by a translation 1 unit left and 5 units down can be combined to form a single translation 5 units left and 2 units down. 4 units left + 1 unit left = 5 units left 3 units up + 5 units down = 2 units down Two or more translations are equivalent to a single translation.

Transformation shape sorter The object of this activity is to use a combination of transformations to fit each shape into its matching hole in as few moves as possible. Reflections can be through any horizontal of vertical line, rotations can be 90°, 180° or 270° around a chosen point and translations can be a given number of units right (negative for left) or up (negative for down).

TRANSFORMATIONS CHANGE THE POSTION OF A SHAPE CHANGE THE SIZE OF A SHAPE TRANSLATION ROTATION REFLECTION DILATION Change in location Turn around a point Flip over a line Change size of a shape