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Mathematics

Quarter 1 – Module 6

Illustrating a Rectangular

Coordinate System

Mathematics – Grade 8 Alternative Delivery Mode Quarter 1 – Module 6 Illustrating a Rectangular Coordinate System First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Printed in the Philippines by ________________________ Department of Education – Caraga Region Office Address: Learning Resource Management Section (LRMS) J.P. Rosales Avenue, Butuan City, Philippines 8600 Tel. No./Telefax No.: (085) 342-8207 / (085) 342- 5969 E-mail Address: [email protected] Development Team of the Module Writers: Jayson Karl D. Dumas, Vincent Butch S. Embolode, Emmanuel S. Saga Language Editor: Merjorie G. Dalagan Layout Evaluator : Jake D. Fraga Content Evaluator : Alsie Mae M. Perolino Reviewers: Rhea J. Yparraguirre , Nilo B. Montaño, Lilibeth S. Apat, Liwayway J. Lubang, Rhodora C. Luga, Lee C. Apas, Jenny O. Pendica, Illustrator: Jayson Karl D. Dumas, Vincent Butch S. Embolode Layout Artist: Jayson Karl D. Dumas, Vincent Butch S. Embolode, Emmanuel S. Saga Management Team: Francis Cesar B. Bringas Isidro M. Biol, Jr. Maripaz F. Magno Josephine Chonie M. Obseñares Josita B. Carmen Celsa A. Casa Regina Euann A. Puerto Bryan L. Arreo Elnie Anthony P. Barcena Leopardo P. Cortes

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Introductory Message

For the facilitator:

Welcome to the Mathematics 8 Alternative Delivery Mode (ADM) Module on Illustrating a

Rectangular Coordinate System!

This module was collaboratively designed, developed and reviewed by educators both

from public and private institutions to assist you, the teacher or facilitator in helping the

learners meet the standards set by the K to 12 Curriculum while overcoming their

personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent

learning activities at their own pace and time. Furthermore, this also aims to help learners

acquire the needed 21st century skills while taking into consideration their needs and

circumstances.

As a facilitator, you are expected to orient the learners on how to use this module. You

also need to keep track of the learners' progress while allowing them to manage their own

learning. Furthermore, you are expected to encourage and assist the learners as they do

the tasks included in the module.

For the learner:

Welcome to the Mathematics 8 Alternative Delivery Mode (ADM) Module on Illustrating a

Rectangular Coordinate System!

This module was designed to provide you with fun and meaningful opportunities for

guided and independent learning at your own pace and time. You will be enabled to

process the contents of the learning resource while being an active learner.

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This module has the following parts and corresponding icons:

What I Need to Know This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know This part includes an activity that aims to check

what you already know about the lesson to take.

If you get all the answers correct (100%), you

may decide to skip this module.

What’s In This is a brief drill or review to help you link the

current lesson with the previous one.

What’s New In this portion, the new lesson will be introduced

to you in various ways; a story, a song, a poem,

a problem opener, an activity or a situation.

What is It This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More This comprises activities for independent practice

to solidify your understanding and skills of the

topic. You may check the answers to the

exercises using the Answer Key at the end of the

module.

What I Have Learned This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Can Do This section provides an activity which will help

you transfer your new knowledge or skill into real

life situations or concerns.

Assessment This is a task which aims to evaluate your level of

mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to you

to enrich your knowledge or skill of the lesson

learned.

Answer Key This contains answers to all activities in the

module.

What I Need to Know

This module was designed and written for you to answer the activity you’ve missed while

you are away from school. It is here to help you master rectangular coordinate system

and its uses. The scope of this module permits it to be used in many different learning

situations. The language used recognizes your diversity and diverse vocabulary level.

The lessons are arranged to follow the standard sequence of the course. But the order in

which you read them can be changed to correspond with the textbook you are now using.

This module contains:

Lesson 1- The Rectangular Coordinate System

After going through this module, you are expected to:

1. define the Rectangular Coordinate System and other related terms;

2. plot the point on a coordinate plane;

3. give the coordinates of a given point on a coordinate plane; and

4. use the coordinate plane to solve problems in daily life.

What I Know

PRE-ASSESSMENT

Choose the letter of the correct answer and write it on your answer sheet.

1. What is a Rectangular Coordinate System?

A. It is a coordinate system that is used for naming points in a plane.

B. It is a coordinate system that is used for graphing linear functions.

C. It is a coordinate system that is used to determine the location of a point by

using a single number.

D. It is a coordinate system that is composed of two perpendicular number lines

that meet at a point of origin.

2. How many quadrants does a Rectangular Coordinate System have?

A. 2 C. 6

B. 4 D. 8

3. Which among these mathematicians was the Cartesian Plane named after?

A. Euclid C. Blaise Pascal

B. Pythagoras D. Rene Descartes

4. What do you call the vertical number line in the Cartesian plane?

A. y-axis C. origin

B. x-axis D. quadrant

5. For the coordinates of the point in quadrant IV, the x value and y value are

always _____ and ______, respectively.

A. negative, positive C. positive, negative

B. negative, negative D. positive, positive

6. The coordinates of the origin in the coordinate plane are________.

A. (0,1) C. (0,0)

B. (1,0) D. (1,1)

7. Point (3, 4) is an example of _______.

A. labels C. ordered pair

B. fraction D. point of origin

Historically, maps played a vital role for travelers and explorers. This map contains vertical and horizontal lines called longitude and latitude, respectively. In this modern day, map applications and the Global Positioning System (GPS) in your mobile phone still utilize the use of horizontal and vertical lines to give you the exact location or coordinate of the place you are looking for. In this lesson, you will learn the concept of Rectangular Coordinate System, plotting points, and locating coordinates which may help you in understanding maps, distance, economics, research and other daily activity. What’s In Activity: PLOT ME! Plot the given point in the number line. Write your answer in a separate sheet of paper. A. 0 B. 3 C. −𝟏 Lesson 1 The Rectangular Coordinate System

D. 0 E. −𝟐 F. 𝟏

Questions:

1. How were you able to locate a positive point and a negative point in the horizontal number line?
2. How were you able to locate a positive point and a negative point in the vertical number line?

Remember:

 The number associated with a point on the number line is called the coordinate of that point.  The coordinate of the origin is zero.  The coordinates of the points to the right of the origin on a horizontal number line and above the origin on a vertical number line form the set of positive integers.  The coordinates of the points to the left of the origin on a horizontal number line and below the origin on a vertical number line form the set of negative integers.

Figure 1 The horizontal number line is called the 𝒙 − 𝒂𝒙𝒊𝒔. The vertical number line is called the 𝒚 − 𝒂𝒙𝒊𝒔. The point of intersection of the horizontal and vertical number lines is called the origin. Each point in the plane can be located using an ordered pair of numbers (𝑥, 𝑦), where 𝑥 is the horizontal distance and 𝑦 is the vertical distance of the point from the origin. The numbers in the ordered pair are called coordinates. The 𝑥 − 𝑣𝑎𝑙𝑢𝑒 of the coordinates (𝑥, 𝑦) of a point is also known as the abscissa , while the 𝑦 − 𝑣𝑎𝑙𝑢𝑒 is known as the ordinate. x - axis y - axis origin

A (𝒙, 𝒚)

The signs of the first and second coordinates of a point vary in the four quadrants as indicated below. This means that you can easily tell which quadrant an ordered pair is located by just simply looking at the signs of the coordinates. There are also points which lie in 𝑡ℎ𝑒 𝑥 − 𝑎𝑛𝑑 𝑦 − 𝑎𝑥𝑒𝑠. The points which lie in the 𝑥 − 𝑎𝑥𝑖𝑠 have coordinates (𝑥, 0 ) and the points which lie in the 𝑦 − 𝑎𝑥𝑖𝑠 have coordinates ( 0 , 𝑦), where 𝑥 and 𝑦 are real numbers. Let us explore the following examples below. Example 1: The points A(0,1), B(2,1), C(0,3), D(-4,2), E(-2,-3), and F(4,-4) can be plotted in the Cartesian plane as shown in the illustration in Figure 2 where:  point A is along the x-axis;  point B is in Quadrant I;  point C is along the y-axis;  point D is in Quadrant II;  point E is in Quadrant III; and  point F is in Quadrant IV. Quadrant I ( +, +) Quadrant II ( −, +) Quadrant III ( −, −) Quadrant IV ( +, −) If both the x- and y- coordinates are positive, then the point is in Quadrant I. If the x- coordinate is negative, and the y- coordinate is positive, then the point is in Quadrant II. If the x- coordinate is positive and the y- coordinate is negative, then the point is in Quadrant IV. If both the x- and y- coordinates are negative, then the point is in Quadrant III.

Answer:  Point M is in Quadrant II. It is located 2 units to the left of the 𝒚 − 𝒂𝒙𝒊𝒔 and 3 units above the 𝒙 − 𝒂𝒙𝒊𝒔. Hence, the coordinates of the point M is (−𝟐, 𝟑).  Point A is in Quadrant III. It is located 1 unit to the left of the 𝒚 − 𝒂𝒙𝒊𝒔 and 2 units below the 𝒙 − 𝒂𝒙𝒊𝒔. Hence, the coordinates of the point A is (−𝟏, −𝟐).  Point T is in Quadrant IV. It is located 2 units to the right of the 𝒚 − 𝒂𝒙𝒊𝒔 and 4 units below the 𝒙 − 𝒂𝒙𝒊𝒔. Hence, the coordinates of the point T is (𝟐, −𝟒).  Point H is in Quadrant I. The point is located 3 units to the right of the 𝒚 − 𝒂𝒙𝒊𝒔 and 2 units above the 𝒙 − 𝒂𝒙𝒊𝒔. Hence, the coordinates of the point H is (𝟑, 𝟐). Example 3. Plot the points on the Cartesian plane and determine the quadrant. Connect each pair of consecutive points and find the perimeter of the resulting quadrilateral. a) L ( 3 , 4 ) b) O (− 3 , 4 ) c) V (− 3 , − 4 ) d) E ( 3 , − 4 ) Answers: Referring to the Cartesian plane in Figure 4 that follows,  L ( 3 , 4 ) means that the point is located 3 units to the right of the y-axis and 4 units above the x – axis. Since the signs of the coordinates are both positive, point L is in Quadrant I.  O (− 3 , 4 ) means that the point is located 3 units to the left of the y-axis and 4 units above the x-axis. Since the sign of the x-coordinate or the abscissa is negative and the sign of the y-coordinate or the ordinate is positive, then point O is in Quadrant II.  V (− 3 , − 4 ) means that the point is located 3 units to the left of the y-axis and 4 units below the x-axis. Since the signs of both x-and y-coordinates are both negative, point V is in Quadrant III.  E ( 3 , − 4 ) means that the point is located 3 units to the right of the y-axis and 4 units below the x-axis. Since the sign of the x-coordinate or abscissa is positive and the sign of the y-coordinate or ordinate is negative, point E is in Quadrant IV. Figure 4

𝑽(−𝟑, −𝟒) 𝑬^ (𝟑,^ −𝟒)

Connecting the adjacent vertices, we see that point L, O, V, and E forms a rectangle. To find the perimeter of the rectangle, we know that: 𝑃 = 2 𝐿 + 2 𝑊 Note that each interval in the Cartesian plane represents one unit of measure. This means that quadrilateral formed by the points L, O, V, E has length of 8 units and width of 6 units. Hence, the perimeter of quadrilateral LOVE is: 𝑃 = 2 𝐿 + 2 𝑊 𝑃 = 2 ( 8 ) + 2 ( 6 ) 𝑃 = 16 + 12 𝑃 = 28 𝑢𝑛𝑖𝑡𝑠 What’s More Activity 1 : Wow Caraga! Describe the location of each point that represents a place in Caraga Region by the completing the following table. An example is done for you. Write your answer in a separate sheet of paper. Place COORDINATES QUADRANT/AXIS Example: Dinagat Islands (-1,3) QII

1. Agusan del Norte
2. Cabadbaran City
3. Surigao City
4. Agusan del Sur
5. Bislig City
6. Surigao del Norte
7. Tandag City

Activity 3 : Plot the Points This activity will enable you to plot the points in the Cartesian Plane given its coordinates. Plot the following points in the Cartesian Plane then identify which quadrant or axis it belongs. Write your answer in a separate sheet of paper.

1. A(− 8 , 4 ) 6. F( 6 , 0 )
2. B(− 2 , − 6 ) 7. G(− 4 , − 4 )
3. C( 5 , 5 ) 8. H( 0 , 0 )
4. D( 0 , − 8 ) 9. I( 5 , 1 )
5. E( 10 , − 3 ) 10. J( 9 , − 7 ) NOTE: When plotting a point in the Cartesian Plane, remember that the first number is for the horizontal axis and the second number is for the vertical axis. Therefore, make the first move either right or left in the x-axis then up or down in the y-axis.

What I Have Learned Fill in the blanks of the appropriate element that would make the sentence correct. Write your answer in a separate sheet of paper.

1. The Rectangular Coordinate System is also termed as _______________ in honor of the French mathematician ______________ who is known as the “Father of Modern Mathematics.”
2. The Rectangular Coordinate Plane is composed of two perpendicular number lines that meet at the intersection called ______ and divide the plane into four regions called ________.
3. In an ordered pair, the first number is the x- coordinate which is also known as the __________ and the second number is the y – coordinate which is also known as __________.
4. The point falls in Quadrant I if it has signs (,), Quadrant II if (,), Quadrant III if (,) and Quadrant IV if (,).
5. For more ease in plotting the points in the Cartesian Plane, start making move from the origin either ____ or ______ in the x-axis then move _____ or ______ in the y-axis.