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**Coordinates and Design**

Chapter One Coordinates and Design

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**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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**Parts of a Cartesian Plane**

The horizontal axis is called the x-axis. The vertical axis is called the y-axis.

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**Quadrants The Coordinate Grid is made up of 4 Quadrants. QUADRANT I**

QUADRANT II QUADRANT III QUADRANT IV

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**Signs of the Quadrants The signs of the quadrants are either**

positive (+) or negative (-). QUADRANT I QUADRANT II QUADRANT III QUADRANT IV (+, +) (-, +) (-, -) (+, -)

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**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.

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**1.1 The Cartesian Plane . Identify Points on a Coordinate Grid .**

B: (5, 3) C: (9, 3) D: (9, 7)

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**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)

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**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

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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

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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?

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**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

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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.

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**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.

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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.

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**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.

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**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).

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Student Name: 10 Vertices

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**Review Through Assignment**

BLM 1-3, BLM 1-4, BLM 1-5, BLM 1-6

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

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**1.3 Transformations Transformations: Translation Reflection Rotation**

Include translations, reflections, and rotations. Translation Reflection Rotation

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**Translation Translations are SLIDES!!!**

Let's examine some translations related to coordinate geometry.

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**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)

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**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)

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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?

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**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

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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.

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**1.3 Transformations: Let’s Practice Textbook Page 25**

Question #4a, b, 5,

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**1.3 Translations 4 a) What is the translation shown in this picture?**

6 units right, 5 units up Or (+6,+5)

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**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)

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**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)

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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.

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**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.

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**Reflecting over the y-axis:**

Where do you think this picture will end up?

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**Reflections Assignment Page 25 Lets go over #7 and #8 as a class.**

Page 26 #10,11, and12 on your own!

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Reflection Question #10

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

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Reflection Question #12

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

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**1.3 Transformations Rotation:**

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

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**Transformations Assignment Page 27**

Lets go over #13 and #14 as a class. Page # 15,16,17, and18 on your own!

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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

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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.

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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.

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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).

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

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**Geometric Transformations**

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

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**Reflection, Rotation, or Translation**

1. Reflection, Rotation, or Translation Rotation

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**Reflection, Rotation, or Translation**

1. Reflection, Rotation, or Translation Rotation

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**Reflection, Rotation, or Translation**

2. Reflection, Rotation, or Translation Translation

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**Reflection, Rotation, or Translation**

2. Reflection, Rotation, or Translation Translation

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**Reflection, Rotation, or Translation**

3. Reflection, Rotation, or Translation Reflection

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**Reflection, Rotation, or Translation**

3. Reflection, Rotation, or Translation REFLECTION Reflection

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**Reflection, Rotation, or Translation**

4. Reflection, Rotation, or Translation Reflection

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**Reflection, Rotation, or Translation**

4. Reflection, Rotation, or Translation Reflection REFLECTION

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**Reflection, Rotation, or Translation**

5. Reflection, Rotation, or Translation Rotation

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**Reflection, Rotation, or Translation**

5. Reflection, Rotation, or Translation Rotation ROTATION

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**Reflection, Rotation, or Translation**

7. Reflection, Rotation, or Translation Reflection

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**Reflection, Rotation, or Translation**

7. Reflection, Rotation, or Translation Reflection

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**Reflection, Rotation, or Translation**

6. Reflection, Rotation, or Translation Translation

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**TRANSLATION – MOVE FROM ONE POINT TO ANOTHER**

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

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**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.

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**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.

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**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.

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**Reflection, Rotation, or Translation**

12. Reflection, Rotation, or Translation Translation. TRANSLATION

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**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°

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**Reflection, Rotation, or Translation**

14. Reflection, Rotation, or Translation ROTATION Rotation

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**Reflection, Rotation, or Translation**

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

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**Reflection, Rotation, or Translation**

16. Reflection, Rotation, or Translation Rotation ROTATION

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**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.

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**Reflection, Rotation, or Translation**

18. Reflection, Rotation, or Translation Translation.

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**Reflection, Rotation, or Translation**

19. Reflection in multiple mirrors. Reflection in multiple mirrors.

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**Reflection, Rotation, or Translation**

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

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**Reflection, Rotation, or Translation**

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

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**Reflection, Rotation, or Translation**

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

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**Transformations Assignment**

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

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**The Ultimate PowerPoint Game**

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**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

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**BattleGraph Example Keep this board HIDDEN from the other team!**

This is the INSIDE board.

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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

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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

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**BattleGraph Example Keep this board VISIBLE!**

This is the OUTSIDE board. Use this board to ATTACK.

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**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

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**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

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**Use this board to ATTACK.**

Keep this board VISIBLE! This is the OUTSIDE board.

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