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Chapter One

Chapter One. Coordinates and Design. 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).

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Chapter One

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  1. 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 II QUADRANT I QUADRANT III QUADRANT IV

  5. Signs of the Quadrants • The signs of the quadrants are either positive (+) or negative (-). QUADRANT II QUADRANT I (+, +) (-, +) 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 A: (5, 7) 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 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.

  15. 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.

  16. 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.

  17. 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).

  18. Student Name: 10 Vertices

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

  20. 1.3 TRANSFORMATIONS This section will focus on the use of Translations, Reflections, Rotations, and describe the image resulting from a transformation.

  21. 1.3 Transformations • Transformations: • Include translations, reflections, and rotations. • TranslationReflection Rotation

  22. Translation • Translations are SLIDES!!! Let's examine some translations related to coordinate geometry.

  23. 1.3 Transformations • Translation: • A slide along a straight line • Count the number of horizontal units and vertical units represented by the translation arrow. • The horizontal distance is 8 units to the right, and the vertical distance is 2 units down • (+8 -2)

  24. 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)

  25. 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?

  26. 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

  27. 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.

  28. 1.3 Transformations: • Let’s Practice • Textbook Page 25 • Question #4a, b, 5,

  29. 1.3 Translations • 4 a) What is the translation shown in this picture? 6 units right, 5 units up Or (+6,+5)

  30. 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)

  31. 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)

  32. 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.

  33. 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. 

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

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

  36. Reflection • Question #10

  37. Reflection • Question #11 The coordinates of A'B'C'D'E'F'G'H' are: • A’(+2,+2) • E'(+2, –4), • G'(+3, –2), • B’(0,+2) • F'(+3, –4), • H'(+2, –2). • C’(-4,-5) • D’(+2, –5),

  38. Reflection • Question #12

  39. TransformationsRotation:A turn about a fixed point called “the center of rotation”The rotation can be clockwise or counterclockwise

  40. 1.3 Transformations • Rotation: • A turn about a fixed point called “the center of rotation” • The rotation can be clockwise or counterclockwise.

  41. Transformations • Assignment • Page 27 • Lets go over #13 and #14 as a class. • Page 27-28 # 15,16,17, and18 on your own!

  42. 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.

  43. 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.

  44. 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.

  45. 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).

  46. Homework Questions • # 18 b) The rotation image is identical to the original image.

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