Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Second Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Conic Sections
Previous Lesson: Binomial Expansion: nth Term and Applications.
Topic: CONIC SECTION: THE CIRCLE
Subject Matter: Meaning of circle (set of points equidistant from a fixed centre), equation of circle given centre and radius, circle as conic section concept

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define a circle as a locus of points.
Explain the concept of a circle as a conic section.
State the standard and general equations of a circle.
Identify the centre and radius from the equation of a circle.

Affective Domain:

Appreciate the geometric representation of a circle.
Develop interest in solving problems related to circles.

Psychomotor Domain:

Derive the equation of a circle given its centre and radius.
Solve problems involving finding the equation of a circle under different conditions.
Determine the centre and radius of a circle from its general equation.

Collaborate with peers to solve problems on circle equations.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics for Senior Secondary Schools by P.N. Okeke and others.
Chart depicting circle as a section of a cone.

Instructional Materials

The teacher will teach this lesson with the aid of:

A chart depicting a circle as a section of a cone.
Whiteboard/chalkboard and markers/chalk.
Further Mathematics textbooks.
Graph papers.

Rationale for the Lesson

This lesson helps pupils understand the fundamental properties and equations of a circle, which is an important geometric shape. This knowledge is essential for further studies in mathematics and for applications in various fields like engineering and physics.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic geometry, coordinate geometry, distance formula, and algebraic manipulation of equations.

Lesson Content/Board Summary

CONIC SECTION: THE CIRCLE

Introduction to Conic Sections

Conic sections are curves formed by the intersection of a plane with a double-napped cone. The type of curve formed depends on the angle at which the plane intersects the cone. Common conic sections include circles, ellipses, parabolas, and hyperbolas.

A circle is formed when a plane intersects a cone perpendicular to its axis. It is a special case of an ellipse.

Definition of a Circle

A circle is defined as the locus of all points in a plane that are equidistant from a fixed point called the centre. The fixed distance from the centre to any point on the circle is called the radius (r).

Equation of a Circle with Centre (h, k) and Radius r (Standard Form)

Let C(h, k) be the centre of the circle and P(x, y) be any point on the circumference. The distance between C and P is the radius, r.

Using the distance formula:
r = √[(x – h)² + (y – k)²]

Squaring both sides gives the standard equation of a circle:

(x – h)² + (y – k)² = r²

If the centre of the circle is at the origin (0, 0), the equation becomes:

x² + y² = r²

Worked Example 1: Find the equation of a circle with centre (2, -3) and radius 5 units.

Step 1: Identify the centre (h, k) and radius r.

h = 2, k = -3, r = 5

Step 2: Substitute these values into the standard equation: (x – h)² + (y – k)² = r².

(x – 2)² + (y – (-3))² = 5²

Step 3: Simplify the equation.

(x – 2)² + (y + 3)² = 25

Worked Example 2: The endpoints of a diameter of a circle are A(-1, 4) and B(5, 2). Find the equation of the circle.

Step 1: Find the centre (h, k) of the circle using the midpoint formula.

h = (x₁ + x₂)/2 = (-1 + 5)/2 = 4/2 = 2

k = (y₁ + y₂)/2 = (4 + 2)/2 = 6/2 = 3

So, the centre C is (2, 3).

Step 2: Find the radius (r) of the circle using the distance formula between the centre and one endpoint (e.g., A).

r = √[(x – h)² + (y – k)²]

r = √[(-1 – 2)² + (4 – 3)²]

r = √[(-3)² + (1)²]

r = √[9 + 1] = √10

Step 3: Substitute the centre (h, k) and radius (r) into the standard equation.

(x – 2)² + (y – 3)² = (√10)²

(x – 2)² + (y – 3)² = 10

General Equation of a Circle

The general equation of a circle is given by:

x² + y² + 2gx + 2fy + c = 0

From this general form, the centre of the circle is (-g, -f) and the radius is r = √(g² + f² – c).

Note: For the radius to be real, g² + f² – c must be greater than or equal to zero.

Worked Example 3: Find the centre and radius of the circle given by the equation x² + y² – 6x + 4y – 12 = 0.

Step 1: Compare the given equation with the general equation x² + y² + 2gx + 2fy + c = 0.

2g = -6 ⇒ g = -3

2f = 4 ⇒ f = 2

c = -12

Step 2: Determine the centre (-g, -f).

Centre = (-(-3), -(2)) = (3, -2)

Step 3: Calculate the radius using r = √(g² + f² – c).

r = √[(-3)² + (2)² – (-12)]

r = √[9 + 4 + 12]

r = √25

r = 5 units

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and asks them to identify some circular objects in the classroom or around them (e.g., clock face, wheel, plate). The teacher then introduces the topic: “Conic Section: The Circle”.
Pupils’ Activity: Pupils identify circular objects and listen attentively to the introduction.
Learning Point: Pupils are engaged and relate to the topic through real-world examples.

Step 2: Introduction to Conic Sections and Circle Formation

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher uses the chart to explain what conic sections are and how a circle is formed when a plane cuts a double-napped cone perpendicular to its axis. The teacher explains that a circle is a special type of conic section.
Pupils’ Activity: Pupils observe the chart, listen to the explanation, and ask questions for clarification.
Learning Point: Pupils understand the origin of a circle as a conic section.

Step 3: Definition of a Circle

Time: 5 minutes
Teaching Skill: Questioning/Discussion
Teacher’s Activity: The teacher guides pupils to define a circle as the locus of points equidistant from a fixed point (centre). The teacher also defines the radius.
Pupils’ Activity: Pupils contribute to the definition and copy the definition into their notebooks.
Learning Point: Pupils understand the geometric definition of a circle.

Step 4: Derivation and Standard Equation of a Circle

Time: 8 minutes
Teaching Skill: Explanation/Derivation
Teacher’s Activity: The teacher explains how to derive the standard equation of a circle (x – h)² + (y – k)² = r² using the distance formula, with the centre at (h, k) and radius r. The teacher also shows the equation for a circle with its centre at the origin.
Pupils’ Activity: Pupils pay attention, ask questions, and copy the derivation and standard equations into their notes.
Learning Point: Pupils learn how the standard equation of a circle is obtained and its forms.

Step 5: Worked Examples on Standard Equation

Time: 8 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher solves Worked Example 1 and Worked Example 2 from the board summary, explaining each step clearly. The teacher encourages pupils to attempt similar problems.
Pupils’ Activity: Pupils follow the steps, ask questions, and attempt to solve the examples as guided by the teacher.
Learning Point: Pupils practice applying the standard equation to solve problems.

Step 6: General Equation of a Circle

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces the general equation of a circle, x² + y² + 2gx + 2fy + c = 0, and explains how to determine the centre (-g, -f) and radius r = √(g² + f² – c) from it. The teacher then solves Worked Example 3.
Pupils’ Activity: Pupils listen, observe the worked example, and copy the general equation and formulas.
Learning Point: Pupils understand the general form of the circle’s equation and how to extract its properties.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define a circle.
How is a circle formed as a conic section?
State the standard equation of a circle with centre (h, k) and radius r.
Find the equation of a circle with centre (-1, 5) and radius 3 units.
Determine the centre and radius of the circle given by the equation x² + y² + 8x – 2y – 8 = 0.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the definition of a circle, its formation as a conic section, and the standard and general forms of its equation. The teacher assigns homework involving more problems on finding circle equations.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

Circle – A set of points equidistant from a fixed point (centre).
Conic Section – A curve formed by the intersection of a plane with a cone.
Radius – The distance from the centre to any point on the circle.
Centre – The fixed point from which all points on the circle are equidistant.
Equation – A mathematical statement showing the relationship between variables.
Standard form – The equation of a circle given as (x – h)² + (y – k)² = r².
General form – The equation of a circle given as x² + y² + 2gx + 2fy + c = 0.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more challenging problems involving tangents or intersections of circles. For those who need more support, the teacher will provide additional guided practice with simpler examples and step-by-step instructions, using visual aids more frequently.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of coordinate geometry, especially the distance and midpoint formulas, as these are foundational for understanding circle equations. Encourage pupils to draw diagrams for each problem to visualize the given information. Emphasize the derivation of the standard equation to foster deeper understanding rather than just memorization. Provide ample practice examples for both standard and general forms of the circle’s equation.