Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 2nd Term
Week: 3
Age: 15 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: General Mathematics
Previous Lesson: Mensuration of Solid Shapes.
Topic: QUADRATIC EQUATIONS (III)
Subject Matter: Plotting graph with a quadratic function and a linear function, using an already plotted curve to find solutions of equations, finding the maximum and minimum values of y with corresponding x, solving quadratic and linear equation graphically, word problems leading to quadratic equations.

Specific Objectives

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

Cognitive Domain:

Define quadratic and linear functions.
Construct tables of values for quadratic and linear functions.
Identify the roots of a quadratic equation from its graph.
Explain how to find the maximum or minimum value of a quadratic function from its graph.

Affective Domain:

Appreciate the importance of graphical methods in solving real-world problems.
Develop a systematic approach to solving problems involving graphs.

Psychomotor Domain:

Draw accurate graphs of quadratic and linear functions.
Read solutions, maximum, and minimum values accurately from graphs.
Solve word problems by translating them into graphical equations and finding solutions.

Collaborate with peers to construct and interpret graphs.
Share ideas and methods for solving graphical problems.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum (Senior Secondary Education)
State Unified Scheme of Work for Senior Secondary Schools
General Mathematics for Senior Secondary Schools by A.O. Salami et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

Graph boards
Graph books
Pencils
Rulers
Erasers

Rationale for the Lesson

This lesson helps pupils understand how to visually solve quadratic and linear equations. It enables them to see the relationships between equations and their graphs, which is important for understanding various real-life situations involving curves and lines, such as projectile motion or cost analysis.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of plotting linear graphs, identifying coordinates, and basic understanding of quadratic equations.

Lesson Content/Board Summary

QUADRATIC EQUATIONS (III): Graphical Solutions

1. Plotting Graphs of Quadratic and Linear Functions

To plot a graph for a quadratic function (e.g., y = ax² + bx + c) or a linear function (e.g., y = mx + c), follow these steps:

Construct a table of values for x and y.
Draw and label the x and y axes on graph paper.
Choose an appropriate scale for both axes.
Plot the points from the table of values.
Draw a smooth curve through the points for a quadratic function or a straight line for a linear function.

2. Using an Already Plotted Curve to Find Solutions of Equations

Finding roots of a quadratic equation:

The roots (solutions) of a quadratic equation (ax² + bx + c = 0) are the x-values where the graph of y = ax² + bx + c intersects the x-axis (where y = 0).

Solving quadratic and linear equations graphically:

To solve a quadratic equation (e.g., ax² + bx + c = 0) and a linear equation (e.g., y = mx + k) simultaneously, plot both graphs on the same axes. The solutions are the coordinates (x, y) of the points where the two graphs intersect.

3. Finding Maximum and Minimum Values of y with Corresponding x

A quadratic graph (parabola) has a turning point. This point represents either the maximum or minimum value of y.

If the parabola opens upwards (U-shape), the turning point is a minimum point.
If the parabola opens downwards (inverted U-shape), the turning point is a maximum point.

The maximum or minimum value of y is the y-coordinate of the turning point, and the corresponding x is the x-coordinate of the turning point.

4. Solving Word Problems Leading to Quadratic Equations Graphically

Word problems can often be translated into quadratic equations. To solve them graphically:

Formulate the quadratic equation from the problem.
Plot the graph of the quadratic function.
Identify the relevant x-intercepts or intersection points that represent the solution to the problem.

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 basic quadratic equations and asks pupils what they know about graphs. The teacher then introduces the topic: Graphical Solutions of Quadratic Equations, explaining that graphs provide a visual way to find solutions.
Pupils’ Activity: Pupils respond to questions about previous knowledge and listen attentively to the introduction.
Learning Point: Pupils recall prior knowledge and are introduced to the lesson topic.

Step 2: Plotting Graphs of Quadratic and Linear Functions

Time: 10 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher demonstrates how to construct a table of values for a simple quadratic function (e.g., y = x² – 4) and a linear function (e.g., y = x + 2). The teacher guides pupils on how to choose an appropriate scale, draw axes, and plot points accurately on a graph board.
Pupils’ Activity: Pupils follow the teacher’s demonstration, construct their own tables of values, and begin plotting points in their graph books.
Learning Point: Pupils learn the practical steps for plotting quadratic and linear graphs.

Step 3: Using an Already Plotted Curve to Find Solutions of Equations

Time: 10 minutes
Teaching Skill: Guidance/Observation
Teacher’s Activity: Using the plotted quadratic graph from Step 2, the teacher guides pupils to observe where the curve crosses the x-axis to find the roots of the equation y = x² – 4 = 0. The teacher then superimposes the linear graph and guides pupils to identify the intersection points to solve the quadratic and linear equations simultaneously.
Pupils’ Activity: Pupils identify the roots and intersection points from the graphs they have drawn, reading the corresponding x and y values.
Learning Point: Pupils understand how to read solutions to quadratic and simultaneous equations from graphs.

Step 4: Finding Maximum and Minimum Values

Time: 5 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains the concept of maximum and minimum values on a quadratic graph. Using the plotted quadratic curve, the teacher points out the turning point and demonstrates how to read the maximum or minimum value of y and its corresponding x-value.
Pupils’ Activity: Pupils identify the turning point on their graphs and determine whether it represents a maximum or minimum value, reading the coordinates.
Learning Point: Pupils learn to identify and interpret maximum and minimum values from a quadratic graph.

Step 5: Solving Quadratic and Linear Equations Graphically

Time: 5 minutes
Teaching Skill: Application/Practice
Teacher’s Activity: The teacher presents another example of solving a quadratic equation (e.g., x² – 2x – 3 = 0) and a linear equation (e.g., y = x – 1) graphically. The teacher provides a brief guided practice for pupils to perform the steps.
Pupils’ Activity: Pupils attempt to solve the given equations graphically in their graph books, applying the skills learned.
Learning Point: Pupils practice solving equations by graphical methods.

Step 6: Word Problems Leading to Quadratic Equations

Time: 5 minutes
Teaching Skill: Problem-solving/Guidance
Teacher’s Activity: The teacher introduces a simple word problem that can be translated into a quadratic equation. The teacher guides pupils on how to formulate the equation and suggests how its solution can be found graphically by plotting the corresponding function.
Pupils’ Activity: Pupils participate in formulating the equation and discuss how to approach the graphical solution.
Learning Point: Pupils understand how to apply graphical solutions to real-world problems.

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 quadratic function and a linear function.
List the steps involved in plotting the graph of a quadratic function.
How do you find the roots of a quadratic equation from its graph?
Explain how to determine the maximum or minimum value of y from a quadratic graph.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, emphasizing the importance of accuracy in plotting graphs and reading values. The teacher assigns homework involving plotting a quadratic graph and finding its roots, maximum/minimum value, and solving it simultaneously with a linear graph.
Pupils’ Activity: Pupils listen to the summary, ask any remaining questions, and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and are given tasks to reinforce learning.

Lesson Keywords

Quadratic equation – An equation of degree 2, typically in the form ax² + bx + c = 0.
Linear equation – An equation of degree 1, typically in the form y = mx + c.
Graph – A diagram showing the relationship between two or more variables, usually as lines or curves.
Root – The solution(s) to an equation, often represented as the x-intercepts on a graph.
Solution – The value(s) that make an equation true.
Maximum value – The highest point (y-coordinate) on a downward-opening quadratic graph.
Minimum value – The lowest point (y-coordinate) on an upward-opening quadratic graph.
Turning point – The point on a quadratic graph where the curve changes direction (maximum or minimum).

Differentiation

For pupils who grasp concepts quickly, they can be given more complex quadratic and linear functions to plot, or word problems requiring more steps. For pupils needing more support, the teacher can provide pre-drawn axes or pre-calculated tables of values, offering individual guidance during plotting and reading from graphs.

Note for teachers using this lesson plan

Ensure that pupils have sufficient graph paper and drawing instruments. Emphasize the importance of clear labeling of axes and accurate plotting of points. Encourage pupils to use a sharp pencil and ruler for precision. Practical demonstration on a large graph board is highly recommended.