Lesson Note on Simultaneous Linear and Quadratic Equations: Elimination, Substitution and Graphs for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 10
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: Quadratic Equation: Word Problems and Real-Life Applications.
Topic: SIMULTANEOUS LINEAR AND QUADRATIC EQUATION (I)
Subject Matter: Revision of simultaneous linear equations, simultaneous linear and quadratic equation by elimination method, simultaneous linear and quadratic equations by substitution method, graphical method, word problem leading to simultaneous linear and quadratic equation, revision of linear and quadratic graph, simultaneous linear and quadratic equations by graphical method

Specific Objectives

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

Cognitive Domain:

• Recall methods for solving simultaneous linear equations.
• Define simultaneous linear and quadratic equations.
• Explain how to solve simultaneous linear and quadratic equations using the substitution method.
• Outline the steps for solving simultaneous linear and quadratic equations graphically.

Affective Domain:

• Appreciate the importance of solving simultaneous equations in real-world problems.
• Develop interest in applying algebraic methods to solve complex problems.
• Show patience and accuracy when solving equations.

Psychomotor Domain:

• Solve simultaneous linear equations using elimination, substitution, and graphical methods.
• Construct tables of values for quadratic equations.
• Draw linear and quadratic graphs accurately.
• Use graphs to find solutions to simultaneous linear and quadratic equations.

Social Domain:

• Participate actively in class discussions on problem-solving techniques.
• Collaborate with peers to solve practice problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Graph board
• Graph books
• Mathematical sets
• Charts showing examples of linear and quadratic graphs

Rationale for the Lesson

This lesson helps pupils understand how to solve different types of equations simultaneously, which is fundamental for advanced mathematics. It enables them to apply these skills to solve practical problems encountered in various fields such as science and engineering.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of solving linear equations, basic algebraic manipulation, and plotting points on a graph.

Lesson Content/Board Summary

Simultaneous Linear and Quadratic Equations

Revision of Simultaneous Linear Equations

Simultaneous linear equations are a set of two or more linear equations that contain two or more variables. The solution to a system of simultaneous linear equations is the set of values for the variables that satisfy all equations in the system.

Methods for solving simultaneous linear equations include:

• Elimination Method: Involves adding or subtracting the equations to eliminate one variable.
• Substitution Method: Involves expressing one variable in terms of the other from one equation and substituting it into the second equation.
• Graphical Method: Involves plotting the graphs of both equations and finding their point of intersection.

Simultaneous Linear and Quadratic Equations by Substitution Method

A simultaneous linear and quadratic equation involves one linear equation and one quadratic equation. The solution(s) are the values of the variables that satisfy both equations.

To solve by substitution:

1. Make one variable the subject of the formula in the linear equation.
2. Substitute this expression into the quadratic equation.
3. Solve the resulting quadratic equation for the remaining variable.
4. Substitute the value(s) found back into the linear equation to find the corresponding value(s) of the other variable.

Example 1: Solve the simultaneous equations:

y = x + 1 (Equation 1)

y = x² – 3x + 4 (Equation 2)

Step 1: Substitute Equation 1 into Equation 2:

x + 1 = x² – 3x + 4

Step 2: Rearrange into a standard quadratic equation (ax² + bx + c = 0):

x² – 3x – x + 4 – 1 = 0

x² – 4x + 3 = 0

Step 3: Solve the quadratic equation by factorization:

(x – 1)(x – 3) = 0

So, x = 1 or x = 3

Step 4: Substitute the values of x back into Equation 1 (y = x + 1) to find y:

When x = 1, y = 1 + 1 = 2

When x = 3, y = 3 + 1 = 4

Solution: The solutions are (x=1, y=2) and (x=3, y=4).

Example 2: Solve the simultaneous equations:

x + y = 7 (Equation 1)

x² + y² = 25 (Equation 2)

Step 1: From Equation 1, make y the subject:

y = 7 – x (Equation 3)

Step 2: Substitute Equation 3 into Equation 2:

x² + (7 – x)² = 25

x² + (49 – 14x + x²) = 25

Step 3: Simplify and rearrange into a quadratic equation:

2x² – 14x + 49 – 25 = 0

2x² – 14x + 24 = 0

Divide by 2: x² – 7x + 12 = 0

Step 4: Solve the quadratic equation by factorization:

(x – 3)(x – 4) = 0

So, x = 3 or x = 4

Step 5: Substitute the values of x back into Equation 3 (y = 7 – x) to find y:

When x = 3, y = 7 – 3 = 4

When x = 4, y = 7 – 4 = 3

Solution: The solutions are (x=3, y=4) and (x=4, y=3).

Simultaneous Linear and Quadratic Equations by Graphical Method

The graphical method involves plotting the graphs of both the linear and quadratic equations on the same coordinate plane. The points where the graphs intersect are the solutions to the simultaneous equations.

Steps:

1. Construct a table of values for the linear equation and plot its straight line graph.
2. Construct a table of values for the quadratic equation (typically for x from -3 to 3 or -4 to 4, depending on the equation) and plot its parabolic curve.
3. Identify the point(s) of intersection of the line and the curve. These points represent the solutions (x, y) to the simultaneous equations.

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 previous knowledge by asking pupils about methods for solving simultaneous linear equations.
Pupils’ Activity: Pupils respond by mentioning elimination, substitution, and graphical methods.
Learning Point: Pupils recall prior knowledge of solving simultaneous linear equations.

Step 2: Revision of Simultaneous Linear Equations

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher guides pupils to solve a simple simultaneous linear equation using either elimination or substitution method as a quick revision.
Pupils’ Activity: Pupils actively participate and solve the revision problem.
Learning Point: Pupils refresh their understanding of simultaneous linear equations.

Step 3: Introduction to Simultaneous Linear and Quadratic Equations by Substitution

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces simultaneous linear and quadratic equations and explains the substitution method, detailing each step.
Pupils’ Activity: Pupils listen attentively and ask questions for clarification.
Learning Point: Pupils understand the concept and steps for the substitution method.

Step 4: Worked Example (Substitution Method)

Time: 7 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher works through Example 1 from the board summary, explaining each step clearly on the board.
Pupils’ Activity: Pupils copy the example and follow the teacher’s steps.
Learning Point: Pupils learn how to apply the substitution method to solve problems.

Step 5: Introduction to Graphical Method

Time: 6 minutes
Teaching Skill: Explanation/Visualisation
Teacher’s Activity: The teacher explains the graphical method for solving simultaneous linear and quadratic equations, emphasizing the need for tables of values and identifying intersection points.
Pupils’ Activity: Pupils observe the charts and listen to the explanation.
Learning Point: Pupils understand the principles of the graphical method.

Step 6: Worked Example (Graphical Method)

Time: 5 minutes
Teaching Skill: Demonstration/Guidance
Teacher’s Activity: The teacher outlines the steps for constructing tables of values for both equations and how to draw the graphs to find the solutions, possibly using a pre-drawn graph.
Pupils’ Activity: Pupils take notes on the steps and observe the graphical representation.
Learning Point: Pupils grasp the procedure for solving graphically.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

1. What are the three methods for solving simultaneous linear equations?
2. Explain the first step when solving a simultaneous linear and quadratic equation using the substitution method.
3. If the graphs of a linear and a quadratic equation intersect at two points, what does this mean for the solutions?
4. Solve the following equations using the substitution method: y = 2x + 1 and y = x² – 2.

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/Assignment
Teacher’s Activity: The teacher summarizes the key learning points of the lesson and assigns homework for further practice.
Pupils’ Activity: Pupils copy the summary and the assigned homework.
Learning Point: Pupils consolidate their learning and prepare for independent practice.

Lesson Keywords

• Simultaneous – Occurring or existing at the same time.
• Linear Equation – An equation whose graph is a straight line.
• Quadratic Equation – An equation of degree two, whose graph is a parabola.
• Substitution Method – An algebraic method for solving simultaneous equations by replacing a variable with an equivalent expression.
• Graphical Method – A method for solving simultaneous equations by plotting their graphs and finding points of intersection.

Differentiation

For pupils who grasp the concept quickly, the teacher will provide more complex word problems leading to simultaneous linear and quadratic equations. For pupils who are struggling, the teacher will provide additional guided practice with simpler examples and offer peer tutoring opportunities.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in basic algebra and graphing before introducing simultaneous linear and quadratic equations. Emphasize step-by-step problem-solving and encourage pupils to check their solutions. Provide ample practice opportunities for both algebraic and graphical methods.