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Starter. Convert the following: 4000 m = __________km 20 mm = __________cm 100 cm = __________ m 45 cm = __________ m 5 km = __________ m. 4. 2. 1. 0.45. 5000. ÷ 10. ÷ 1000. ÷ 100. km. cm. m. mm. × 10. × 100. × 1000. Geometry. Transformations.

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  1. Starter • Convert the following: • 4000 m = __________km • 20 mm = __________cm • 100 cm = __________ m • 45 cm = __________ m • 5 km = __________ m 4 2 1 0.45 5000 ÷ 10 ÷ 1000 ÷ 100 km cm m mm × 10 × 100 × 1000

  2. Geometry Transformations

  3. 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

  4. 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 2-3-2 seating plan in economy class?

  5. Symmetry

  6. Axes of Symmetry 2 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:

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

  8. 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)

  9. Rotational order of Symmetry 6 2 4 8 3 5

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

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

  12. 4 8 4 0 2 2 1 1 2 6 6 12

  13. 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 ?

  14. 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)

  15. 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.

  16. Reflect across the y-axis A A’ B B’ C’ C E’ E D D’

  17. Reflect A’B’C’D’E’ across the line y=x A’ Which are the invariant points? B’ D’’ and E’’ C’ E’ A’’ E’’ D’ D’’ C’’ B’’

  18. 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. IGCSE Ex 3 pg 279-280 Ex 4 pg 280-281

  19. Analyze the ALPHABET ALPHABET 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?

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

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

  22. Rotation

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

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

  25. 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 0o 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.

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

  27. Rotation • In rotationevery 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 it’s image are always the same distance from the centre of rotation. • The centre of rotation is the only invariant point. • Rotation game

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

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

  30. By what angle is this flag rotated about point C ? C 270º IGCSE Ex 5 pg 282-283 Ex 6 pg 283-284

  31. Define these terms The line equidistant from an object and its image The point an object is rotated about Doesn’t change • Mirror line • Centre of rotation • Invariant What is invariant in • Reflection • rotation 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

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

  33. Vectors • Vectors describe movement ( ) x ← movement in the x direction (left and right) y ← movement in the y direction (up and down) Each vertex of shape EFGH moves along the vector ( ) -3 -6 To become the translated shape E’F’G’H’

  34. Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`. ( ) -4 - 2 IGCSE Ex 7 pg 285-280

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

  36. To draw an enlargement • Measure the distance from the centre of enlargement to a point. • Multiply the point by the scale factor and mark the point’s image point. • Continue for as many points as necessary.

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

  38. To find the centre of enlargement • Join each of the points to it’s image point. • The point where all lines intersect is the centre of enlargement.

  39. Calculating the scale factor To calculate the scale factor (k) we use the formula : Scale factor (k) = = = IGCSE Ex 8pg 287

  40. 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: • Measure the distance from the centre of rotation to a point. • Multiply the distance by the scale factor. • Measure the distance on the opposite side of the centre of rotation from the point. • Repeat for as many points as necessary.

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

  42. Do Now: • Match up the terms with the correct definition • Write them into your vocab list 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 Image Mirror line Perpendicular Bisector Reflection

  43. Do Now: What transformations are in each example? 1. 2. 3. Rotation Enlargement Reflection 4. 5. Reflection Translation

  44. Combined Transformations Notice that, like composite functions, we work from right to left Let A be a reflection in line x = 3 Let B be a translation of Let C be a rotation of 180° about O (0,0) A(▲1) means ‘perform the transformation A on triangle ▲1 BA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1 CBA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1, then perform the transformation C

  45. A(▲1) means ‘perform the transformation A on triangle ▲1 ▲1 ▲1 ▲2 Let A be a reflection in line x = 3

  46. BA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1 (▲2) ▲2 ▲1 ▲3 Let B be a translation of

  47. CBA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1, then perform the transformation C ▲1 ▲2 ▲4 ▲3 Let C be a rotation of 180° about O (0,0)

  48. Repeated Transformations CC(▲1) Let C be a rotation of 180° about O (0,0)

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