coordinates and design. the cartesian plane (or coordinate grid) is made up of two directed real...

Chapter OneCoordinates and Design

The Cartesian Plane (or coordinate grid) is made up of two directed real lines that intersect perpendicularly at their respective zero points.

What is a Cartesian Plane

ORIGINThe point where the x-axis and the y-axis cross(0,0)

The horizontal axis is called the x-axis.

The vertical axis is called the y-axis.

Parts of a Cartesian Plane

The Coordinate Grid is made up of 4 Quadrants.

Quadrants

QUADRANT IQUADRANT II

QUADRANT III QUADRANT IV

The signs of the quadrants are either positive (+) or negative (-).

Signs of the Quadrants

QUADRANT IQUADRANT II

QUADRANT III QUADRANT IV

(+, +)(-, +)

(-, -) (+, -)

1.1 The Cartesian Plane

Identify Points on a Coordinate Grid

A: (x, y)

B: (x, y)

C: (x, y)

D: (x, y)

. .

1.1 The Cartesian Plane Identify Points on a Coordinate Grid

A: (5, 7)

B: (5, 3)

C: (9, 3)

D: (9, 7)

When we read coordinates we read them in the order x then y

Plot 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)

What are common mistakes when constructing a Coordinate Plane?

1. Units not the same in terms of intervals2. Switch the order that they appear3. Wrong symbols for quadrants

Textbook: Page 9

#5, 7, for questions 9 and 10 plot on two separate graphs. Graph paper is provided for you.

Challenge #14, 16

Assignment

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?

1.2 Create Designs

What is a vertex?

AB

C

•A vertex is a point where two sides of a figure•meet.•The plural is vertices!

The vertices of the Triangle areA (x, y)B (x, y)C (x, y)

A (4, 4)

B (0, 4)

C (2, 0)

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.

Create Designs

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.

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.

Try This

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.

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).

Student Name:

10 Vertices

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

Review Through Assignment

1.3 TRANSFORMATIONS

This section will focus on the use of Translations, Reflections, Rotations, and

describe the image resulting from a transformation.

1.3 Transformations Transformations:

◦ Include translations, reflections, and rotations. Translation Reflection Rotation

Translations are SLIDES!!!

Translation

Let's examine some translations related to coordinate geometry.

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)

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)

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?

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

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.

1.3 Transformations: Let’s Practice

Textbook Page 25

◦ Question #4a, b, 5,

1.3 Translations 4 a) What is the translation shown

in this picture?

6 units right, 5 units up Or

(+6,+5)

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)

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)

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.

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.

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

Reflections Assignment

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

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

Reflection Question #10

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

◦ A’(+2,+2) B’(0,+2)

C’(-4,-5)

D’(+2, –5),

E'(+2, –4), F'(+3, –4),

G'(+3, –2), H'(+2, –2).

Reflection Question #12

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

1.3 Transformations Rotation:

◦A turn about a fixed point called “the center of rotation”

◦The rotation can be clockwise or counterclockwise.

Transformations Assignment

Page 27 Lets go over #13 and #14 as a class.

Page 27-28 # 15,16,17, and18 on your own!

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.

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.

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.

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).

Homework Questions # 18

b) The rotation image is identical to the original image.

by D. Fisher

Geometric Transformations

REFLECTION

Reflection, Rotation, or Translation

1.


2.


3.


3.

REFLECTION


4.


4.

REFLECTION


5.


5.

ROTATION


7.


6.


8.

TRANSLATION – MOVE FROM ONE POINT TO

ANOTHER

Why is this not perfect reflection?

10.

Why is this not perfect reflection?

10.

ZEBRAS HAVE SLIGHTLY DIFFERENT STRIPING


11.

PROBABLY DOESN’T FIT ANY CATEGORY


12.

TRANSLATION


13.

Why possibly both? Either reflected or

rotated 180°


14.

ROTATION


15.

REFLECTION IN SEVERAL DIRECTIONS


16.

ROTATION


17.


18.


19.

Reflection in multiple mirrors.


20.


21.


22.

Transformations Assignment

Page 36-37 # 1-10, 12, 15, 16, 18 and 21 on your own!

The Ultimate PowerPoint Game

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

BattleGraph Directions

BattleGraph Example

This is the INSIDE board.

Keep this board HIDDEN

from the other team!

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

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

BattleGraph Example

Use this board to ATTACK.

Keep this board VISIBLE!

This is the OUTSIDE board.

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

Use this board to

HIDE your battleship

s.

Keep this board

HIDDENfrom the

other team!

This is the INSIDE board.

Battleships1 Aircraft Carrier (AAAAA)1 Cruiser

(CCCC)1 Destroyer

(DDD)1 Submarine

(SS)

Home Page

Use this board to ATTACK.

Keep this board VISIBLE!

This is the OUTSIDE board.

coordinates and design. the cartesian plane (or coordinate grid) is made up of two directed real...

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