The Equation of a Circle

 Determine the center and radius of a circle given its equation and vice versa

 Graph a circle and other geometric figures on the coordinate plane

 Solve problems involving geometric figures on the coordinate plane

Writer:

Melvin M. Callanta

Essential

Understanding:

Students will understand that the concepts involving plane coordinate geometry are useful tools in solving real-life problems like finding

Transfer Goal:

Students will be able to apply with perseverance and accuracy the key concepts of plane coordinate

geometry in formulating and solving problems involving geometric figures on the rectangular coordinate plane.

B. Planning for Assessment Product/Performance

The following are products and performances that students are expected to come up with in this module.

1. Ground Plan drawn on a grid with coordinates

2. Equations and problems involving mathematics concepts already learned such as coordinate plane, slope and equation of a line, parallel and perpendicular lines, polygons, distance, angles, etc

3. Finding the distance between a pair of points on the coordinate plane 4. Determining the missing coordinates of the endpoints of a segment 5. Finding the coordinates of the midpoint of the segment whose endpoints

are given

6. Describing the figure formed by a set of points on a coordinate plane 7. Determining the missing coordinates corresponding to the vertices of

some polygons

8. Solutions to problems involving the distance and the midpoint formulas 9. Coordinate Proofs of some geometric properties

10. Sketch of a municipal, city, or provincial map on a coordinate plane with the coordinates of some important landmarks

11. Formulating and solving real-life problems involving the distance and the midpoint formula

12. Finding the radius of a circle drawn on a coordinate plane

13. Determining the center and the radius of a circle given the equation 14. Graphing a circle given the equation

15. Writing the equation of a circle given the center and the radius

16. Writing the equation of a circle from standard form to general form and vice-versa

17. Determining the equation that describes a circle

18. Solutions to problems involving the equation of a circle

19. Formulating and solving real-life problems involving the equation of a circle

200 Assessment Map

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE Pre-Assessment/ radius of a circle given the endpoints of a diameter Finding the center of a circle given the

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE Pre-Test: find the midpoint of a segment

202

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE Writing a center of a circle Explaining how to the center and the radius equation of a circle Summative Post-Test:

Part I

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE Illustrating the radius of a circle given the endpoints of a diameter Finding the center of a circle given the

Self-Assessment Journal Writing:

Expressing understanding of the distance formula, midpoint formula, coordinate proof, and the equation of a circle.

204 Assessment Matrix (Summative Test))

Levels of

Assessment What will I assess? How will I

assess? How Will I Score? formula to prove some geometric properties.

 Illustrate the center-radius form of the equation of a circle.

 Determine the center and radius of a circle given its equation and vice versa. Solving (maximum of 4 points for each

problem)

Product/

Performance 30%

The learner is able to formulate and solve problems involving geometric figures on the rectangular coordinate

Rubric for Grip Map of the Municipality

C. Planning for Teaching-Learning

This module covers key concepts of plane coordinate geometry. It is divided into two lessons, namely: The Distance Formula and the Equation of a Circle.

In Lesson 1 of this module, the students will derive the distance formula and apply it in proving geometric relationships and in solving problems, particularly finding the distance between objects or points. They will also learn about the midpoint formula and its applications. Moreover, the students will graph and describe geometric figures on the coordinate plane.

The second lesson is about the equation of a circle. In this lesson, the students will illustrate the center-radius form of the equation of a circle, determine the center and the radius given its equation and vice-versa, and show its graph on the coordinate plane (or by using the computer freeware, GeoGebra). More importantly, the students will solve problems involving the equation of a circle.

In learning the equation of a circle, the students will use their prior knowledge and skills through the different activities provided. This is to connect and relate those mathematics concepts and skills that students previously studied to their new lesson. They will also perform varied learning tasks to process the knowledge and skills learned and to further deepen and transfer their understanding of the different lessons in real-life situations.

Introduce the main lesson to the students by showing them the pictures below, then ask them the questions that follow:

206

Entice the students to find the answers to these questions and to determine the vast applications of plane coordinate geometry through this module.

Objectives:

After the learners have gone through the lessons contained in this module, they are expected to:

1. derive the distance formula;

2. find the distance between points;

3. determine the coordinates of the midpoint of a segment;

4. name the missing coordinates of the vertices of some geometric figures;

5. write a coordinate proof to prove some geometric relationships;

6. give/write the center-radius form of the equation of a circle;

7. determine the center and radius of a circle given its equation and vice versa;

8. write the equation of a circle from standard form to general form and vice versa;

9. graph a circle and other geometric figures on the coordinate plane; and 10. solve problems involving geometric figures on the coordinate plane.

Look around! What geometric figures do you see in your classroom, school buildings, houses, bridges, roads, and other structures? Have you ever asked yourself how geometric figures helped in planning the construction of these structures?

In your community or province, was there any instance when a stranger or a tourist asked you about the location of a place or a landmark? Were you able to give the right direction and its distance? If not, could you give the right information the next time somebody asks you the same question?

PRE-ASSESSMENT:

Assess students’ prior knowledge, skills, and understanding of mathematics concepts related to the Distance Formula, the Midpoint Formula, the Coordinate Proof, and the Equation of a Circle. These will facilitate teaching and students’ understanding of the lessons in this module.

LEARNING GOALS AND TARGETS:

Students are expected to demonstrate understanding of key concepts of plane coordinate geometry, formulate real-life problems involving these concepts, and solve these with perseverance and accuracy.

Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate