Presentation on theme: "Coordinates and Design"— Presentation transcript:
1 Coordinates and Design Chapter OneCoordinates and Design
2 What is a Cartesian Plane The Cartesian Plane (or coordinate grid) is made up of two directed real lines that intersect perpendicularly at their respective zero points.ORIGINThe point where the x-axis and the y-axis cross(0,0)
3 Parts of a Cartesian Plane The horizontal axis is called the x-axis.The vertical axis is called the y-axis.
4 Quadrants The Coordinate Grid is made up of 4 Quadrants. QUADRANT I QUADRANT IIQUADRANT IIIQUADRANT IV
5 Signs of the Quadrants The signs of the quadrants are either positive (+) or negative (-).QUADRANT IQUADRANT IIQUADRANT IIIQUADRANT IV(+, +)(-, +)(-, -)(+, -)
6 1.1 The Cartesian Plane Identify Points on a Coordinate Grid A: (x, y) B: (x, y)C: (x, y)D: (x, y)HINT: To find theX coordinate count howmany units to the rightif positive,or how many units to theleft if negative.
7 1.1 The Cartesian Plane . Identify Points on a Coordinate Grid . B: (5, 3)C: (9, 3)D: (9, 7)
8 When we read coordinates we read them in the order x then yPlot the following pointson the smart boardA: (9, -2)B: (7, -5)C: (2, -4)D: (2, -1)E: (0, 1)F: (-2, 3)G: (-7, 4)
9 What are common mistakes when constructing a Coordinate Plane? Units not the same in terms of intervalsSwitch the order that they appearWrong symbols for quadrants
10 AssignmentTextbook: Page 9 #5, 7, for questions 9 and 10 plot on two separate graphs. Graph paper is provided for you. Challenge #14, 16
11 1.2 Create DesignsPut your thinking cap on! What is the following question asking us to find?Label each vertex of each shape.Question!What is a vertex?
12 1.2 Create Designs A vertex is a point where two sides of a figure What is a vertex?A vertex is a point where two sides of a figuremeet.The plural is vertices!The vertices of the Triangle areA (x, y)B (x, y)C (x, y)BAA (4, 4)B (0, 4)C (2, 0)C
13 Create DesignsGraphic Artists use coordinate grids to help them make certain designs. Flags, corporate logos can all be constructed through the use of our coordinate grids.
14 1.2 Create Designs Study the following Flag. How many vertices can you find in the design.Imagine seeing this on a coordinate grid.Notice how it is centered and equally distributed on each side.
19 1.2 Create Designs Assignment: You have been hired to create a flag for the company “Flags R Us!” They are looking for a new creative design that can be based on an interest or hobby of yours. The flag design can be a cool pattern or related to any sport, hobby, or activity you are involved with.The flag needs to have a minimum of 10 Vertices.They want a detailed location of any 10 vertices located on the bottom of your design (list the coordinates).It is your responsibility to use a coordinate grid to create your own pattern.
20 Evaluation Your Flag will be evaluated as following” Neatness: (Have you made sure to color inside the lines).Vertices: (Do you have at least 10).Design: (Have you used designs and shapes to create an image).Handout: (Do you have all the vertices clearly labeled in a legend).
23 1.3 TRANSFORMATIONSThis section will focus on the use of Translations, Reflections, Rotations, and describe the image resulting from a transformation.
24 1.3 Transformations Transformations: Translation Reflection Rotation Include translations, reflections, and rotations.Translation Reflection Rotation
25 Translation Translations are SLIDES!!! Let's examine some translations related to coordinate geometry.
26 1.3 Transformations Translation: A slide along a straight line Count the number of horizontal units and vertical units represented by the translation arrow.Label the vertices A, B, CLabel the new translation A’, B’, C’The horizontal distance is 8 units to the right, and the vertical distance is 2 units down(+8 -2)
27 1.3 Transformations Translation: Count the number of horizontal units the image has shifted.Count the number of vertical units the image has shifted.We would say theTransformation is:1 unit left,6 units upor(-1+,6)
28 In this example we have moved each vertex 6 units along a straight line. If you have noticed the corresponding A is now labeled A’What about the other letters?
29 A translation "slides" an object a fixed distance in a given direction 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 in the same direction
30 When you are sliding down a water slide, you are experiencing a translation. Your body is moving a given distance (the length of the slide) in a given direction. You do not change your size, shape or the direction in which you are facing.
32 1.3 Translations 4 a) What is the translation shown in this picture? 6 units right, 5 units upOr(+6,+5)
33 Horizontal Distance is: 1.3 Translations4 b) What is the translation in the diagram below?Horizontal Distance is:6 units leftVertical Distance is:4 units upOr(-6,+4)
34 1.3 Translations #5 B) The coordinates of the translation image are P'(+7, +4), Q’(+7, –2),R'(+6, +1), S'(+5, +2).C) The translation arrow isshown: 3 units right and6 units down. (+3, -6)
35 ReflectionsIs figure A’B’C’D’ a reflection image of figure ABCD in the line of reflection, n?How do you know?Figure A'B'C'D' IS a reflection image of figure ABCD in the line of reflection, n.Each vertex in the red figure is the same distance from the line of reflection, n, as its reflected vertex in the blue image.
36 A reflection is often called a flip A reflection is often called a flip. Under a reflection, the figure does not change size.It is simply flipped over the line of reflection.Reflecting over the x-axis:When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.
43 Transformations Rotation: A turn about a fixed point called “the center of rotation” The rotation can be clockwise or counterclockwise
44 1.3 Transformations Rotation: A turn about a fixed point called “the center of rotation”The rotation can be clockwise or counterclockwise.
45 Transformations Assignment Page 27 Lets go over #13 and #14 as a class.Page # 15,16,17, and18 on your own!
46 1.3 TransformationsPg 27. #13a) The coordinates for ∆HAT are H(–3, –2), A(–1, –3), and T(–3, –6).The coordinates for ∆HAT are H(–3, –2), A(–1, –3), and T(–3, –6).b) The rotation is 180 counterclockwise.Discuss the different angles of rotation:90, 190, 270, 360
47 RotationsPg 27 #15.a) The coordinates for the centre of rotation are (–4, –4).b) Rotating the figure 90° clockwise will produce the same image as rotating it 270° in the opposite direction, or counterclockwise.
48 Homework Questions#16a) The coordinates for the centre of rotation are (+2, –1).b) The direction and angle of the rotation could be 180° clockwise or 180° counterclockwise.
49 Homework Questions#17a) The figure represents the parallelogram rotated about C, 270° clockwise.b) The coordinates for Q'R'S'T' are Q'(–1, –1), R'(–1, +2), S'(+1, +1), and T'(+1, –2).
50 Homework Questions# 18b) The rotation image is identical to the original image.
51 Geometric Transformations REFLECTIONPlease send feedback to Answers and discussion are in the notes for each slide.by D. Fisher
52 Reflection, Rotation, or Translation 1.Reflection, Rotation, or TranslationRotation
53 Reflection, Rotation, or Translation 1.Reflection, Rotation, or TranslationRotation
54 Reflection, Rotation, or Translation 2.Reflection, Rotation, or TranslationTranslation
55 Reflection, Rotation, or Translation 2.Reflection, Rotation, or TranslationTranslation
56 Reflection, Rotation, or Translation 3.Reflection, Rotation, or TranslationReflection
57 Reflection, Rotation, or Translation 3.Reflection, Rotation, or TranslationREFLECTIONReflection
58 Reflection, Rotation, or Translation 4.Reflection, Rotation, or TranslationReflection
59 Reflection, Rotation, or Translation 4.Reflection, Rotation, or TranslationReflectionREFLECTION
60 Reflection, Rotation, or Translation 5.Reflection, Rotation, or TranslationRotation
61 Reflection, Rotation, or Translation 5.Reflection, Rotation, or TranslationRotationROTATION
62 Reflection, Rotation, or Translation 7.Reflection, Rotation, or TranslationReflection
63 Reflection, Rotation, or Translation 7.Reflection, Rotation, or TranslationReflection
64 Reflection, Rotation, or Translation 6.Reflection, Rotation, or TranslationTranslation
65 TRANSLATION – MOVE FROM ONE POINT TO ANOTHER 8.Reflection, Rotation, or TranslationTRANSLATION – MOVE FROM ONE POINT TO ANOTHERTranslation
66 Why is this not perfect reflection? 10.Why is this not perfect reflection?The zebras have slightly different striping. One has its nose closer to the ground.
67 Why is this not perfect reflection? 10.Why is this not perfect reflection?ZEBRAS HAVE SLIGHTLY DIFFERENT STRIPINGThe zebras have slightly different striping. One has its nose closer to the ground.
68 Reflection, Rotation, or Translation 11.Reflection, Rotation, or TranslationPROBABLY DOESN’T FIT ANY CATEGORYReflection is probably the best answer because the inside part of the bird’s foot is slightly shorter than the outside part. However, this example from nature does not really fit exactly in any of the categories.
69 Reflection, Rotation, or Translation 12.Reflection, Rotation, or TranslationTranslation.TRANSLATION
70 Reflection, Rotation, or Translation 13.Reflection, Rotation, or TranslationReflection. However, rotation of 180o will be the same.Why possibly both?Either reflected or rotated 180°
71 Reflection, Rotation, or Translation 14.Reflection, Rotation, or TranslationROTATIONRotation
72 Reflection, Rotation, or Translation 15.Reflection, Rotation, or TranslationREFLECTION IN SEVERAL DIRECTIONSReflection in several directions.
73 Reflection, Rotation, or Translation 16.Reflection, Rotation, or TranslationRotationROTATION
74 Reflection, Rotation, or Translation 17.Reflection, Rotation, or TranslationReflection. Note the position of the purple tips; rotation of 180o would cause the top purple tip to be on the bottom.
75 Reflection, Rotation, or Translation 18.Reflection, Rotation, or TranslationTranslation.
76 Reflection, Rotation, or Translation 19.Reflection in multiple mirrors.Reflection in multiple mirrors.
77 Reflection, Rotation, or Translation 20.Reflection, Rotation, or TranslationTranslation. Watch the colors.
78 Reflection, Rotation, or Translation 21.Reflection, Rotation, or TranslationReflection. Note the position of the red parts.
79 Reflection, Rotation, or Translation 22.Reflection, Rotation, or TranslationRotation. Note the red parts.
80 Transformations Assignment Page # 1-10, 12, 15, 16, 18 and 21 on your own!
82 BattleGraph Directions Each team will hide their 4 battleships in their HIDDEN Mathematical Ocean by writing the correct number of points for each battleship with its corresponding letterAll ships must be either horizontal or verticalShips may not overlapDraw a rectangle around the correct number of points for each battleship
83 BattleGraph Example Keep this board HIDDEN from the other team! This is the INSIDE board.
84 ATTACKERS & DEFENDERSTeams will take turns being the ATTACKERS and the DEFENDERSThe ATTACKERS will select a place to attack by giving an ordered pair of numbers to the DEFENDERSThe ATTACKERS will then write the ordered pair in the box to the side and circle that point on their VISIBLE Mathematical OceanThe DEFENDERS will find the coordinate on their HIDDEN Mathematical Ocean and circle itThe DEFENDERS will say if the attack was a HIT (ATTACKERS fill-in circle) or a MISS (ATTACKERS leave circle empty)Teams will then switch roles
85 Winning BattleGraphIf the coordinate is not written in the box on the side, the attack is automatically a MISSIf the coordinate is not in the Mathematical Ocean, the attack is automatically a MISSIf the ATTACKERS sink one of your battleships, you must tell tell them. Otherwise you will LOSE one turn.The ATTACKERS will connect the points once the entire ship is SUNK.To WIN the game you must sink all of the the other team’s battleships before they sink all of yours
86 BattleGraph Example Keep this board VISIBLE! This is the OUTSIDE board.Use this board to ATTACK.
87 Get Ready to Hide Your Battleships Aircraft Carrier(5 A points)Cruiser(4 C points)Destroyer(3 D points)Submarine(2 S points)on the HIDDEN Mathematical Ocean
88 Battleships Use this board to HIDE your battleships. 1 Aircraft Carrier(AAAAA)1 Cruiser(CCCC)1 Destroyer(DDD)1 Submarine(SS)Use this board to HIDE your battleships.Keep this board HIDDEN from the other team!This is the INSIDE board.Home Page
89 Use this board to ATTACK. Keep this board VISIBLE!This is the OUTSIDE board.