1. Coordinates and Design
Chapter One
Coordinates 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.
ORIGIN
The 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 II
QUADRANT III
QUADRANT IV

5. Signs of the Quadrants The signs of the quadrants are either
positive (+) or negative (-).
QUADRANT I
QUADRANT II
QUADRANT III
QUADRANT 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 the
X coordinate count how
many units to the right
if positive,
or how many units to the
left 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 y
Plot the following points
on the smart board
A: (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 intervals
Switch the order that they appear
Wrong symbols for quadrants

10. Assignment
Textbook: 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 Designs
Put 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 figure
meet.
The plural is vertices!
The vertices of the Triangle are
A (x, y)
B (x, y)
C (x, y) B A
A (4, 4)
B (0, 4)
C (2, 0) C

13. Create Designs
Graphic 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.

17. Try This
On page 12 of your text an assignment is given to draw a flag. Plot the points with the proper labels and color the inside of the flag design red. Graph paper is supplied to you.

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).

21. Student Name:
10 Vertices

22. Review Through Assignment
BLM 1-3, BLM 1-4, BLM 1-5, BLM 1-6

23. 1.3 TRANSFORMATIONS
This 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, C
Label 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 the
Transformation is:
1 unit left,6 units up or
(-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.

31. 1.3 Transformations: Let’s Practice Textbook Page 25
Question #4a, b, 5,

32. 1.3 Translations 4 a) What is the translation shown in this picture?
6 units right, 5 units up Or
(+6,+5)

33. Horizontal Distance is:
1.3 Translations
4 b) What is the translation in the diagram below?
Horizontal Distance is:
6 units left
Vertical Distance is:
4 units up Or
(-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 is
shown: 3 units right and
6 units down. (+3, -6)

35. Reflections
Is 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. 

38. Reflecting over the y-axis:
Where do you think this picture will end up?

39. Reflections Assignment Page 25 Lets go over #7 and #8 as a class.
Page 26 #10,11, and12 on your own!

40. Reflection
Question #10

41. Reflection Question #11 The coordinates of A'B'C'D'E'F'G'H' are:

42. Reflection
Question #12

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 Transformations
Pg 27. #13
a) 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. Rotations
Pg 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
#16
a) 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
#17
a) 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
# 18
b) The rotation image is identical to the original image.

51. Geometric Transformations
REFLECTION
Please send feedback to Answers and discussion are in the notes for each slide.
by D. Fisher

52. Reflection, Rotation, or Translation1.
Reflection, Rotation, or Translation
Rotation

53. Reflection, Rotation, or Translation1.
Reflection, Rotation, or Translation
Rotation

54. Reflection, Rotation, or Translation2.
Reflection, Rotation, or Translation
Translation

55. Reflection, Rotation, or Translation2.
Reflection, Rotation, or Translation
Translation

56. Reflection, Rotation, or Translation3.
Reflection, Rotation, or Translation
Reflection

57. Reflection, Rotation, or Translation3.
Reflection, Rotation, or Translation
REFLECTION
Reflection

58. Reflection, Rotation, or Translation4.
Reflection, Rotation, or Translation
Reflection

59. Reflection, Rotation, or Translation4.
Reflection, Rotation, or Translation
Reflection
REFLECTION

60. Reflection, Rotation, or Translation5.
Reflection, Rotation, or Translation
Rotation

61. Reflection, Rotation, or Translation5.
Reflection, Rotation, or Translation
Rotation
ROTATION

62. Reflection, Rotation, or Translation7.
Reflection, Rotation, or Translation
Reflection

63. Reflection, Rotation, or Translation7.
Reflection, Rotation, or Translation
Reflection

64. Reflection, Rotation, or Translation6.
Reflection, Rotation, or Translation
Translation

65. TRANSLATION – MOVE FROM ONE POINT TO ANOTHER8.
Reflection, Rotation, or Translation
TRANSLATION – MOVE FROM ONE POINT TO ANOTHER
Translation

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 STRIPING
The zebras have slightly different striping. One has its nose closer to the ground.

68. Reflection, Rotation, or Translation
11.
Reflection, Rotation, or Translation
PROBABLY DOESN’T FIT ANY CATEGORY
Reflection 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 Translation
Translation.
TRANSLATION

70. Reflection, Rotation, or Translation
13.
Reflection, Rotation, or Translation
Reflection. 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 Translation
ROTATION
Rotation

72. Reflection, Rotation, or Translation
15.
Reflection, Rotation, or Translation
REFLECTION IN SEVERAL DIRECTIONS
Reflection in several directions.

73. Reflection, Rotation, or Translation
16.
Reflection, Rotation, or Translation
Rotation
ROTATION

74. Reflection, Rotation, or Translation
17.
Reflection, Rotation, or Translation
Reflection. 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 Translation
Translation.

76. Reflection, Rotation, or Translation
19.
Reflection in multiple mirrors.
Reflection in multiple mirrors.

77. Reflection, Rotation, or Translation
20.
Reflection, Rotation, or Translation
Translation. Watch the colors.

78. Reflection, Rotation, or Translation
21.
Reflection, Rotation, or Translation
Reflection. Note the position of the red parts.

79. Reflection, Rotation, or Translation
22.
Reflection, Rotation, or Translation
Rotation. Note the red parts.

80. Transformations Assignment
Page # 1-10, 12, 15, 16, 18 and 21 on your own!

81. The Ultimate PowerPoint Game

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 letter
All ships must be either horizontal or vertical
Ships may not overlap
Draw 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 & DEFENDERS
Teams will take turns being the ATTACKERS and the DEFENDERS
The ATTACKERS will select a place to attack by giving an ordered pair of numbers to the DEFENDERS
The ATTACKERS will then write the ordered pair in the box to the side and circle that point on their VISIBLE Mathematical Ocean
The DEFENDERS will find the coordinate on their HIDDEN Mathematical Ocean and circle it
The 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 BattleGraph
If the coordinate is not written in the box on the side, the attack is automatically a MISS
If the coordinate is not in the Mathematical Ocean, the attack is automatically a MISS
If 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.