Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 2
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra
Previous Lesson: Roots of Quadratic Equation: Meaning, Sum and Product of Roots.
Topic: ROOTS OF QUADRATIC EQUATION II
Subject Matter: Conditions for given line to intersect a curve (two real roots, b^2 > 4ac), be tangent to curve (equal roots, b^2 = 4ac), not intersect a curve (no real roots, complex roots, b^2 < 4ac), solution of problems on roots of quadratic equation.

Specific Objectives

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

Cognitive Domain:

State the conditions for a quadratic equation to have equal roots, real roots, and no real roots.
Explain the relationship between the discriminant and the nature of roots.
Solve problems involving the nature of roots of quadratic equations.

Affective Domain:

Appreciate the application of the discriminant in determining the relationship between a line and a curve.
Develop a systematic approach to solving problems related to quadratic roots.

Psychomotor Domain:

Accurately apply the discriminant formula to determine the nature of roots.
Solve problems where conditions for line-curve intersection (tangent, intersection, no intersection) are given.

Collaborate with peers to discuss solutions to problems on quadratic roots.

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.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing conditions for lines to intersect a curve, be tangent to a curve, and not intersect a curve.
Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils understand how the discriminant of a quadratic equation determines the nature of its roots. This knowledge is important for solving various mathematical problems and for understanding the graphical relationship between lines and curves in coordinate geometry.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of solving quadratic equations and calculating the discriminant.

Lesson Content/Board Summary

ROOTS OF QUADRATIC EQUATION II

The Discriminant

For a quadratic equation in the form (ax^2 + bx + c = 0), the discriminant is given by the expression ( Delta = b^2 – 4ac ). The value of the discriminant determines the nature of the roots of the quadratic equation.

Conditions for the Nature of Roots

The nature of the roots of a quadratic equation (ax^2 + bx + c = 0) can be determined by the value of its discriminant ( Delta = b^2 – 4ac ):

If ( Delta > 0 ): The equation has two distinct real roots.
If ( Delta = 0 ): The equation has two equal real roots (or one repeated real root).
If ( Delta < 0 ): The equation has no real roots (it has two complex conjugate roots).

Line-Curve Intersection

When solving for the points of intersection between a line (y = mx + c_1) and a curve (y = ax^2 + bx + c_2), we equate the two equations to get a quadratic equation (ax^2 + (b-m)x + (c_2-c_1) = 0). The discriminant of this resulting quadratic equation determines the relationship between the line and the curve:

If ( Delta > 0 ): The line intersects the curve at two distinct points.
If ( Delta = 0 ): The line is tangent to the curve, intersecting at exactly one point.
If ( Delta < 0 ): The line does not intersect the curve.

Worked Examples

Example 1: Determine the nature of the roots of the equation (3x^2 – 7x + 2 = 0).

Solution:

Step 1: Identify a, b, and c.

For (3x^2 – 7x + 2 = 0), (a = 3), (b = -7), (c = 2).

Step 2: Calculate the discriminant ( Delta = b^2 – 4ac ).

( Delta = (-7)^2 – 4(3)(2) )

( Delta = 49 – 24 )

( Delta = 25 )

Step 3: Interpret the discriminant.

Since ( Delta = 25 > 0 ), the equation has two distinct real roots.

Example 2: Find the value of (k) for which the line (y = x + k) is tangent to the curve (y = x^2 + 3x + 4).

Solution:

Step 1: Equate the line and curve equations to find the intersection points.

(x + k = x^2 + 3x + 4)

Step 2: Rearrange into a standard quadratic equation (ax^2 + bx + c = 0).

(0 = x^2 + 3x – x + 4 – k)

(x^2 + 2x + (4 – k) = 0)

Step 3: Identify a, b, and c for the new quadratic equation.

(a = 1), (b = 2), (c = 4 – k).

Step 4: Apply the condition for tangency.

For the line to be tangent to the curve, the discriminant must be zero (( Delta = b^2 – 4ac = 0 )).

( (2)^2 – 4(1)(4 – k) = 0 )

( 4 – 4(4 – k) = 0 )

( 4 – 16 + 4k = 0 )

( -12 + 4k = 0 )

( 4k = 12 )

Step 5: Solve for (k).

(k = frac{12}{4})

(k = 3)

Therefore, when (k = 3), the line (y = x + 3) is tangent to the curve (y = x^2 + 3x + 4).

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 reviews the previous lesson on solving quadratic equations and the concept of the discriminant. The teacher then introduces the topic by stating that the discriminant can also describe the relationship between a line and a curve.
Pupils’ Activity: Pupils recall previous knowledge on quadratic equations and pay attention to the introduction.
Learning Point: Pupils connect prior knowledge to the new topic.

Step 2: The Discriminant

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines the discriminant ( Delta = b^2 – 4ac ) for a quadratic equation (ax^2 + bx + c = 0) and explains its role in determining the nature of roots.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and formula of the discriminant.

Step 3: Conditions for the Nature of Roots

Time: 8 minutes
Teaching Skill: Explanation/Categorization
Teacher’s Activity: The teacher explains the three conditions for the nature of roots: ( Delta > 0 ) (two distinct real roots), ( Delta = 0 ) (two equal real roots), and ( Delta < 0 ) (no real roots/complex roots). The teacher uses examples to illustrate each condition.
Pupils’ Activity: Pupils listen, copy the conditions, and understand the implications of each discriminant value.
Learning Point: Pupils learn how to use the discriminant to determine the type of roots.

Step 4: Line-Curve Intersection

Time: 8 minutes
Teaching Skill: Connection/Demonstration
Teacher’s Activity: The teacher explains how the discriminant applies to determining if a line intersects a curve at two points, is tangent to the curve (one point), or does not intersect the curve at all. The teacher uses the chart to demonstrate these relationships graphically.
Pupils’ Activity: Pupils observe the chart, listen to the explanation, and understand the practical application of the discriminant to geometry.
Learning Point: Pupils understand the relationship between the discriminant and line-curve interactions.

Step 5: Worked Example 1

Time: 7 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher guides pupils through Example 1 from the board summary, demonstrating how to determine the nature of roots for a given quadratic equation by calculating the discriminant.
Pupils’ Activity: Pupils follow the steps, ask questions, and copy the solution into their notes.
Learning Point: Pupils practice applying the discriminant to find the nature of roots.

Step 6: Worked Example 2

Time: 5 minutes
Teaching Skill: Problem Solving/Application
Teacher’s Activity: The teacher guides pupils through Example 2 from the board summary, focusing on finding an unknown constant when a line is tangent to a curve. The teacher emphasizes setting the discriminant to zero for tangency.
Pupils’ Activity: Pupils actively participate in solving the problem, discussing steps, and verifying the solution.
Learning Point: Pupils learn to solve problems involving conditions for line-curve relationships.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

What is the discriminant of a quadratic equation (ax^2 + bx + c = 0)?
State the condition for a quadratic equation to have two distinct real roots.
When does a quadratic equation have no real roots?
If a line is tangent to a curve, what can be said about the discriminant of the resulting quadratic equation?

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/Assignment
Teacher’s Activity: The teacher summarizes the main points of the lesson, reiterating the importance of the discriminant in determining the nature of roots and line-curve relationships. The teacher assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

Discriminant – The expression (b^2 – 4ac) in a quadratic equation.
Real roots – Solutions to a quadratic equation that are real numbers.
Equal roots – When a quadratic equation has two identical real solutions.
Complex roots – Solutions to a quadratic equation that involve imaginary numbers, occurring when there are no real roots.
Tangent – A line that touches a curve at exactly one point.
Intersect – When a line crosses a curve at one or more points.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving inequalities or finding ranges of values for (k). For pupils needing more support, the teacher can provide additional guided examples and focus on one type of problem at a time.

Note for teachers using this lesson plan

Ensure pupils clearly understand the algebraic manipulation required when equating line and curve equations. Emphasize the importance of setting up the quadratic equation correctly before applying the discriminant. Encourage pupils to sketch graphs for visual understanding, especially for line-curve intersection problems.