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Angles
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Starter Identify 5 right angles in the classroom?
How many right angles are there around a point? How many degrees are in a right angle? How many degrees are there around a point? How many right angles are there on this flag? 16 360 90 4 Door corner, book corners, desk corner……
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Classroom Expectations
Show respect to all other people and property Be on time to class Bring all necessary equipment to class Enter the room sensibly Listen in silence when the teacher or another student is talking Raise your hand when you want to ask or answer a question Work without disturbing others Complete all homework
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Vocabulary Vertex Arm
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C A T We name this angle Ex Question 2,3
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An angle measures the amount a line has to turn to fit onto another line
We measure angles in degrees.
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Vocabulary to do with Angle Sizes
An acute angle is between 0 and 90 degrees
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A right angle is 90 degrees
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An obtuse angle is between 90 and 180 degrees
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A straight angle is 180 degrees
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A reflex angle is between 180 and 360 degrees
Do Exercise go onto if finished
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105° 180° 90° 24° 150° 348° 78° 248° Obtuse Acute Reflex Right Angle Straight
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Obtuse Acute Reflex Right Angle Straight 150° 105° 78° 24° 248° 348° 90° 180°
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We generally measure angle in degrees with a protractor
Exercise do any 5
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Angle Winner Example(1) 45 50 42 Amy 1 2 3 4 5 6 7 8 Jade Estimation
Kate Estimation Exact size Winner Example(1) 45 50 42 Amy 1 2 3 4 5 6 7 8
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Complementary Angles are angles that add to 90°.
55° X=35° x=?
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Adjacent angles on a straight line sum 1800
1200 600 Supplementary angles
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Activity Draw a straight line on your paper. Name each angle of your triangle. One A,B and C Tear, (or cut) the angles (corners) away from your paper and lay them out alone the line. C A B
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Angles in a Triangle Angles in a triangle sum 1800
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Two different objects with acute angles
Four different objects with right angles Two different objects with obtuse angles One straight angle Two different reflex angles around the room Bonus Question Can you find any angles that are complementary or supplementary?
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Angles around a point sum 3600
1200 a 1600 a = 80 degrees
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Vertex Vertically opposite angles Are opposite each other at a vertex
They are equal
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Page 378 and 380 Exercise and 25.10
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Angles on parallel lines
Transversal
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Corresponding angles are congruent
1000 a a = 100 degrees
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Co-interior angles sum 1800
720 g = 108 degrees
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k 700 Alternate angles are congruent K = 70 degrees
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Starter!!! You can choose to do the left hand side or the right hand side. 50º 70º B= A= C= 120º C= A= B= A=____ B=____ C=____ D=____ E=____ 60º 50º 70º 85º 60º E= 30º E= 90º D= D= A=____ B=____ C=____ D=____ E=____ 70º 110º 70º 95º 95º
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Beta Book Page 246 Ex
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Game Rules Have a counter on each of the spots.
The first player can remove 1 or 2 counters off the board (if they take two they must be connecting dots.) The winner is the player who picks up the last counter!!! Play a few games and try to work out a winning strategy
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3D Shapes
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Starter Name the following shapes and give examples of them from your life.
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Face Edges and Vertices Hidden Lines
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Exercises Page 457 Exercise 29.01 Questions 4, 6, 9,14 and 15
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Answers from Text Book 4- a) 4 b) 6 c) 4 6- Draw on the board
9- Next Page 14- Tetrahedron 15- a) Trapeziums b) 12 c) Pyramid Barton, D., Alpha Mathematics Second Edition
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6 12 8 5 9 18 7 15 10 Shape No. Faces Edges Vertices a Cube b
Triangular Prism 5 9 C Hexagonal Prism 18 d Pyramid e Cube where a triangular slice has been cut off. 7 15 10 Barton, D., Alpha Mathematics Second Edition
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Isometric Drawings
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Starter
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Tetracubes A solid made up of four identical cubes joined face to face
There are eight different tetracubes Make each tetracube then draw it in your isometric paper Make sure you don’t just draw the same tetracube from a different angle
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Views Top Right Left Left Front Right 1 1 1 1 Front Plain View
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Right Front Left Draw your own copy of this block formation on your isometric paper Draw the front, left, right and top views Draw the plan view of the shape.
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Front Right Try and make this shape!!! (Then draw it) Left Top
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Front Left Right Top Views Right Left Front Plain View 2 1 3
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Notes if you want them!!! Plain View- shows the height of different parts of the solid when viewed from above. There are also profile views which show the shape from one side only.
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Views Left Top Plain View 2 1 3 Front Right
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Nets A plan that shows all the faces of a solid. It shows how all the faces could be folded and joined to make the shape. Dashed lines show where to fold. Often tabs are included for ease of constructing the shape from the net.
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Instructions Make a net for a closed cube.
How many faces does a closed cube have? What is the smallest number of tabs that you could use so that every joint is secure? What coloured square will be on the opposite side of the cube from the green one?
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Make Your Cubes!!!!!
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Further 3D Shapes
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Closed figures made up of straight sides.
Polygons Closed figures made up of straight sides.
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Examples of Polygons Triangle Quadrilateral Pentagon Hexagon Octagon
Decagon
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Angles of Polygons Interior angles- are angles
between the sides of the polygon on the inside. Exterior angles- are angles found by extending the sides of the polygon.
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Exterior Angles Measure the exterior angles of your polygons.
d Exterior Angles Measure the exterior angles of your polygons. Add the exterior angles of each shape together. What do they add to? Shape Total Degrees of Exterior Angles Triangle Quadrilateral Pentagon Hexagon 360 The sum of the exterior angles of a polygon is 360°.
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Adjacent angles on a straight line
Alternate angles When two parallel lines are cut by a third line, then angles in alternate positions equal in size. Co-interior angles When two parallel lines are cut by a third line, co-interior angles are supplementary. Angles at a point. The sum of the sizes of the angles at a point is 360 Adjacent angles on a straight line The sum of the sizes of the angles on a line is 180 degrease
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Adjacent angles in a right angle
The sum of the size's of the angles in around different points but the same angle Vertically opposite angles Vertically opposite angles are equal in size. Corresponding angles When two parallel lines are cut by a third line, then angles in corresponding positions are equal in size.
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Interior Angles of a Polygon
ttrsgf Interior Angles of a Polygon No. of sides of polygon 3 4 5 You do 6, 8, 10 Drawing Number of triangles 1 2 Degrees in a triangle sum to 180° If there are 180 degrees in a triangle how many degrees must there Be in a quadrilateral which is split into 2 triangles. The rule (n2) × 180° n is the number of sides of the polygon
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Regular Polygons A polygon is called regular if all its sides are the same length and all its angles are the same size. e.g. equilateral triangle, a square or a regular pentagon.
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Exterior Angles Number of sides Sum of exterior angles
Each exterior angle Equilateral triangle Square Pentagon Hexagon Octagon Decagon 3 4 5 6 8 10 360 120 90 72 60 45 36 Interior Angles Number of sides Sum of exterior angles Each exterior angle Equilateral triangle Square Pentagon Hexagon Octagon Decagon 180 360 540 720 1080 1440 60 90 108 120 135 144 3 4 5 6 8 10
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Navigation and Bearings
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Bearings Bearings are angles which are measured clockwise from north. They are always written using 3 digits. The bearings start at 000 facing north and finish at 360 facing north. Bearings 045 120 180 270
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Bearings
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Starter
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Paper Cutting
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Translation A translation is a movement in which each point moves in the same direction by the same distance. To translate an object all you need to know is the image of one point. Every other point moves in the same distance in the same direction.
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A B E’ F’ B’ D’ C’ G’ A' G F E D C
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Reflection In a reflection, and object and its image are on opposite sides of a line of symmetry. This line is often called a mirror line. m
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m
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Where would the mirror line go??
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Invariant Points If a point is already on the mirror line, it stays where it is when reflected. These points are called invariant points.
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m m
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Exercises Today Ex 26.03 2 of 1a, b, c or d. Page397
26.04 Question 2, 7 and 8. Page 401 & 402 Any three questions from Page 404
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Rotation Rotation is a transformation where an object is turned around a point to give its image. Each part of the object is turned through the same angle. To rotate an object you need to know where the center of rotation is and the angle of rotation.
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The angle of rotation This can be given in degrees or as a fraction such as a quarter turn. The direction the object is turned can be either clockwise or anti-clockwise.
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D A C B D’ A’ B’ C’
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Rotation
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What are these equivalent angles of Rotation?
Rotations are always specified in the anti clockwise direction What are these equivalent angles of Rotation? 270º Anti clockwise is _______ clockwise 180º Anti clockwise is ______ clockwise 340º Anti clockwise is _______ clockwise
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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 it’s image are always the same distance from the centre of rotation. The centre of rotation is the only invariant point.
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Drawing Rotations ¼ turn clockwise = B C 90º clockwise A B’ D D’ C’
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Drawing Rotations ¼ Turn anti-clockwise = 90º Anti-clockwise C’ B C A
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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 Centre of rotation
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180º By what angle is this flag rotated about point C ?
Remember: Rotation is always measured in the anti clockwise direction!
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By what angle is this flag rotated about point C ?
90º
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By what angle is this flag rotated about point C ?
270º
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Questions Ex 26.07 Page 411 Qn 1, 2, 3, 7 and 8.
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Answers 26.07 1 D C B’ C’ D’ B A’ A A’ C’ C 2 B’ A B
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C D 3 A B’ A’ B D C D’ C’ B A C’ B’ 7 D’ A’
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s s 8
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Ex a) P b) R c) QS 180° 0° or 360° 8. a) R b) Q c) CB
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Rotational Symmetry A figure has rotational symmetry about a point if there is a rotation other than 360° when the figure can turn onto itself. Order of rotational symmetry is the number of times a figure can map onto itself.
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Order of rotational symmetry=
4 3
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Line Symmetry A shape has line symmetry if it reflects or folds onto itself. The fold is called an axis of symmetry. 2 Line symmetry=
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Total order of symmetry
The number of axes of symmetry plus the order of rotational symmetry. 2+2=4
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-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 1. Move the red dot by the following values and state where it now lies. (-1), (4), (7), (-4) and (11). -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 1. Move the green dot by the following values and state where it now lies. (-6), (3), (-9), (-4) and (5).
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Groups Group One- Kelly, Ruby, Bella, Chrisanna and Bianca
Group Two- Hannah B, Grace, Shanice, Grace and Georgia R Group Three- Hannah C, Remy, Olivia, Kendyl and Sarah Group Four- Kelsey, Shaquille, Kiriana, Cadyne and Claudia Group Five- Lauran, Ashlee, Sophie, Georgia W and Emily S Group Six- Esther, Emily M, Jemma, Amelia and Julia
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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 describe movement + + - - x y
← movement in the x direction (right and left) y ← movement in the y direction (up and down)
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( ) Vectors Vectors describe movement
Each vertex of shape EFGH moves along the vector ( ) -3 -6 To become the translated shape E’F’G’H’
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Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`.
( ) - 4 - 2
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Vectors +3 +4 -5 +4 +2 -6 -6 -2
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