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Starter Convert the following: 4000 m = __________km 20 mm = __________cm 100 cm = __________ m 45 cm = __________ m 5 km = __________ m mm cm m km ÷ 10 ÷ 100 ÷ 1000 × 10× 100×

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Geometry Transformations

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Why is it important that an airplane is symmetrical? Are the freight containers mirror images of each other? How are the blades of the engine symmetrical? Reflection Rotation

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Which aircraft does not have a symmetrical seating plan? Is it possible for a symmetrical aircraft to have an odd number of seats in a row? Which aircraft has a seating plan in economy class?

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Symmetry

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Axes of Symmetry A line of symmetry divides a shape into two parts, where each part is a mirror image of the other half. Example: 2 axes of symmetry

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Line Symmetry - How many axes of Symmetry can you find?

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Note 1: Order of Rotational Symmetry The order of rotational symmetry is how many times the object can be rotated to map itself. (through an angle of 360° or less)

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Rotational order of Symmetry

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Which one of these cards has a Rotational Order of Symmetry = 2?

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Note 1: Total order of symmetry Total order of Symmetry = Number of Axes of Symmetry + Order of Rotational Symmetry (Line Symmetry)

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ShapeAxes of Symmetry Order of Rotational Symmetry Total Order of Symmetry

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Task ! Choose 3 objects in the room and describe their axes of symmetry, order of rotational symmetry and total order of symmetry. Can you find an object with a total order of symmetry greater than 4 ?

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Note 1: Total order of symmetry Total order of symmetry = the number of axes of symmetry + order of rotational symmetry. The number of axes of symmetry is the number of mirror lines that can be drawn on an object. The order of rotational symmetry is how many times the object can be rotated to map itself. (through an angle of 360° or less) IWB Ex pg Ex pg 750 IWB Ex pg Ex pg 750

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Note 2: Reflection A point and its image are always the same distance from the mirror line If a point is on the mirror line, it stays there in the reflection. This is called an invariant point.

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Reflection To draw an image: Measure the perpendicular distance from each point to the mirror line. Measure the same perpendicular distance in the opposite direction from the mirror line to find the image point. (often it is easier to count squares). e.g. Draw the image of PQR in the mirror line LM. IWB Ex pg IWB Ex pg

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Analyze the Notice the letter B, H and E are unchanged if we take their horizontal mirror image? Can you think of any other letters in the alphabet that are unchanged in their reflection? What is the longest word you can spell that is unchanged when placed on a mirror? Can you draw an accurate reflection of your own name? IWB Ex pg IWB Ex pg

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To draw a mirror line between a point and its reflection: 1. Construct the perpendicular bisector between the point and its image. e.g.Find the mirror line by which B` has been reflected from B.

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Practice Drawing a Reflections and mirror lines! Count squares or measure with a ruler Handouts – Reflection, Mirror lines Homework - Finish these handouts.

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Draw the Mirror lines for these shapes using a compass

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What are these equivalent angles of Rotation? 270° Anti clockwise is _______ clockwise 180 ° Anti clockwise is ______ clockwise 340 ° Anti clockwise is _______ clockwise Rotations are always specified in the anti clockwise direction

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Drawing Rotations A B C D B D C ¼ turn clockwise = 90º clockwise Rotate about point A

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To draw images of rotation: Measure the distance from the centre of rotation to a point. Place the protractor on the shape with the cross- hairs on the centre of rotation and the 0 o towards the point. Mark the wanted angle, ensuring to mark it in the anti-clockwise direction. Measure the same distance from the centre of rotation in the new direction. Repeat for as many points as necessary.

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Examples Rotate flag FG, 180 about O Draw the image A`B`C`D` of rectangle ABCD if it is rotated 90 o about point A. A

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Rotation In rotation every point rotates through a certain angle about a fixed point called the centre of rotation. Rotation is always done in an anti-clockwise direction. A point and its image are always the same distance from the centre of rotation. The centre of rotation is the only invariant point. Rotation game

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By what angle is this flag rotated about point C ? 180º Remember: Rotation is always measured in the anti clockwise direction! C

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By what angle is this flag rotated about point C ? 90º C

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By what angle is this flag rotated about point C ? 270º C

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Define these terms Mirror line Centre of rotation Invariant What is invariant in Reflection rotation The line equidistant from an object and its image The point an object is rotated about Doesnt change The mirror line The size of angles and sides The area of the shape Centre of rotation Size of angles and sides The area of the shape

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Translations Each point moves the same distance in the same direction There are no invariant points in a translation (every point moves)

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Vectors Vectors describe movement ( ) x y movement in the x direction (left and right) movement in the y direction (up and down) Each vertex of shape EFGH moves along the vector ( ) To become the translated shape EFGH

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Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`. ( )

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Enlargement In enlargement, all lengths and distances from a point called the centre of enlargement are multiplied by a scale factor (k).

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To draw an enlargement 1.Measure the distance from the centre of enlargement to a point. 2.Multiply the point by the scale factor and mark the points image point. 3.Continue for as many points as necessary.

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Enlarge the ABC by a scale factor of 2 using the point O as the centre of enlargement.

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To find the centre of enlargement 1. Join each of the points to its image point. 2.The point where all lines intersect is the centre of enlargement.

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Calculating the scale factor To calculate the scale factor (k) we use the formula : Scale factor (k) = = =

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Negative Scale Factors When the scale factor is negative, the image is on the opposite side of the centre of enlargement from the object. To draw images of negative scale factors: 1.Measure the distance from the centre of rotation to a point. 2.Multiply the distance by the scale factor. 3.Measure the distance on the opposite side of the centre of rotation from the point. 4.Repeat for as many points as necessary.

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Enlarge XYZ by a scale factor of –2 about O. Y Z X

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Do Now: 1.Match up the terms with the correct definition 2.Write them into your vocab list Image Mirror line Perpendicular Bisector Reflection Cuts a line into two equal parts (cuts it in half) – also called the mediator The transformed object A transformation which maps objects across a mirror line A line which intersects a line at right angles The line in which an object is reflected

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Reflect the shape in the red mirror line, translate the image by the vector Enlarge the image scale factor 3, centre P Rotate 45 o, centre A A P

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Do Now: What transformations are in each example? Reflection Enlargement Rotation Translation Reflection

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Do Now 1.) Reflection is a transformation which maps an object across a __________. 2.) In rotation, the only invariant point is called the __________________. 3.) A _______ describes the movement up and down, and across, in a translation. 4.) All of the _________ points in reflection lie on the mirror line. 5.) The area of the object, the size of the angles and the length of the sides are invariant in both rotation and ___________ mirror line centre of rotation vector invariant reflection invariant, centre of rotation, reflection, vector, mirror line

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Koru Design Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least: One reflection One translation One rotation

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How to Write Instructions Reflect ABCD through the mirror line (M) Translate the image ABCD 4 cm to the right Rotate the image ABCD, 90 o counter clockwise about the point P. M 4 cm P

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Now its your turn! Write the appropriate instruction for each transformation, in the order that it appears. M 5 cm 6 cm A A A A A P

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Writing Instructions 1.) Label your object 2.) Reflect image about mirror line M 3.) Translate the image abcd by the vector ( ) 4.) Rotate the image abcd about point P 90º M a b c d a bc d 1 x -7 d cb a P 3 4 d c a b 4 y 2

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Writing Instructions 1.) Construct equilateral triangle abc (Label your object) 2.) Rotate object abc 90º about point P to give abc. 3.) Reflect image abc through mirror line M to give image abc 4.) Translate the object abc by the vector ( ) to give image abc M a b c a b c c b a P c a b

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Koru Design Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least: One reflection One translation One rotation * Write a set of instructions so another student could reproduce your pattern.

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Choose a Task Design a Logo Using at least 2 different construction techniques and at least 2 different types of transformations with Instructions (You have 20 minutes to complete it!) or The Backyard Follow the set of instructions to complete a plan of a backyard.

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Exchange your Work Logo Creators Follow your partners instructions and recreate their logo Backyard Designers Mark your partners work against the marking sheet

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