Applications of Linear and Quadratic Equations for SS 3

Class: Senior Secondary School 3 (SS3 / SSS 3)
Term: 1st Term
Week: 7 – 8
Age: 17 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: Amortization, Bonds, Debentures, Shares and Taxes
Topic: APPLICATION OF LINEAR AND QUADRATIC EQUATION
Subject Matter: Revision of simultaneous linear and quadratic equations, word problems on linear equations, word problems on simultaneous linear equations, word problems on one linear and one quadratic equation, application to capital market

Specific Objectives

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

Cognitive Domain:

• Revise the methods for solving simultaneous linear and quadratic equations.
• Translate word problems into linear, simultaneous linear, and one linear/one quadratic equations.
• Solve various types of word problems involving linear and quadratic equations.
• Describe the application of linear and quadratic equations in the capital market.

Affective Domain:

• Appreciate the importance of linear and quadratic equations in solving real-life problems.
• Develop an interest in applying mathematical concepts to practical scenarios.
• Exhibit patience and precision when solving complex word problems.

Psychomotor Domain:

• Accurately set up and solve linear and quadratic equations from given word problems.
• Demonstrate the ability to interpret and model real-world situations using algebraic equations.

Social Domain:

• Participate actively in class discussions and problem-solving sessions.
• Collaborate with peers to solve challenging word problems.

Reference Materials

The following resources were used in planning this lesson:

• FCT-ERC SS 3 Scheme of Work, September 2014 curriculum.
• Relevant State Unified Scheme of Work.
• A suitable General Mathematics textbook for SS 3.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Solution charts of simultaneous linear and quadratic equations
• Whiteboard/Blackboard
• Markers/Chalk
• Textbooks

Rationale for the Lesson

This lesson helps students to understand how linear and quadratic equations are used to solve problems in everyday life and various fields. It enables students to translate real-world scenarios into mathematical models, which is an important skill for future studies and careers.

Prerequisite/Previous Knowledge

Students are expected to have prior knowledge of solving linear equations, quadratic equations, and basic algebraic manipulations.

Lesson Content/Board Summary

APPLICATION OF LINEAR AND QUADRATIC EQUATIONS

Revision of Simultaneous Linear Equations

Simultaneous linear equations are a set of two or more linear equations that involve the same variables and are solved together to find a common solution that satisfies all the equations.

Common methods for solving simultaneous linear equations include:

1. Substitution Method: Express one variable in terms of the other from one equation and substitute it into the second equation.
2. Elimination Method: Multiply one or both equations by constants to make the coefficients of one variable equal or additive inverses, then add or subtract the equations to eliminate that variable.

Example 1 (Substitution Method):
Solve:
1) x + y = 7
2) 2x – y = 5

Solution:
Step 1: From equation (1), express y in terms of x: y = 7 – x.
Step 2: Substitute y = 7 – x into equation (2):
2x – (7 – x) = 5
2x – 7 + x = 5
Step 3: Collect like terms and solve for x:
3x – 7 = 5
3x = 5 + 7
3x = 12
x = 12 / 3
x = 4
Step 4: Substitute x = 4 back into y = 7 – x to find y:
y = 7 – 4
y = 3
Therefore, x = 4, y = 3.

Example 2 (Elimination Method):
Solve:
1) 3x + 2y = 13
2) x – 2y = -1

Solution:
Step 1: Notice that the coefficients of y are +2 and -2. Add equation (1) and equation (2) to eliminate y:
(3x + 2y) + (x – 2y) = 13 + (-1)
3x + x + 2y – 2y = 12
4x = 12
Step 2: Solve for x:
x = 12 / 4
x = 3
Step 3: Substitute x = 3 into equation (1) to find y:
3(3) + 2y = 13
9 + 2y = 13
2y = 13 – 9
2y = 4
y = 4 / 2
y = 2
Therefore, x = 3, y = 2.

Revision of Simultaneous Linear and Quadratic Equations

This involves solving a system where one equation is linear and the other is quadratic. The primary method used is substitution.

Example:
Solve:
1) y = x + 1
2) x² + y² = 13

Solution:
Step 1: Substitute equation (1) into equation (2) to eliminate y:
x² + (x + 1)² = 13
Step 2: Expand and simplify the quadratic equation:
x² + (x² + 2x + 1) = 13
2x² + 2x + 1 = 13
2x² + 2x + 1 – 13 = 0
2x² + 2x – 12 = 0
Step 3: Divide by 2 to simplify:
x² + x – 6 = 0
Step 4: Factorize the quadratic equation to find x:
(x + 3)(x – 2) = 0
So, x = -3 or x = 2
Step 5: Substitute the values of x back into equation (1) to find the corresponding y values:
If x = -3: y = -3 + 1 = -2
If x = 2: y = 2 + 1 = 3
Therefore, the solutions are (x, y) = (-3, -2) and (x, y) = (2, 3).

Word Problems on Linear Equations

Word problems require careful reading to translate the given information into a single linear equation with one unknown.

Steps to solve word problems on linear equations:

1. Read the problem carefully to understand the context.
2. Identify the unknown quantity and assign a variable (e.g., x) to it.
3. Formulate a linear equation based on the relationships described in the problem.
4. Solve the equation for the unknown variable.
5. Check your answer to ensure it makes sense in the context of the problem.

Example:
When 5 is added to three times a number, the result is 23. Find the number.

Solution:
Step 1: Let the number be x.
Step 2: Three times the number is 3x.
Step 3: When 5 is added to three times the number, it is 3x + 5.
Step 4: The result is 23, so the equation is: 3x + 5 = 23.
Step 5: Solve for x:
3x = 23 – 5
3x = 18
x = 18 / 3
x = 6
The number is 6.

Word Problems on Simultaneous Linear Equations

These problems involve two unknown quantities and relationships that can be expressed as two linear equations.

Steps to solve word problems on simultaneous linear equations:

1. Read the problem and identify the two unknown quantities.
2. Assign different variables (e.g., x and y) to these unknowns.
3. Formulate two linear equations based on the information provided.
4. Solve the system of equations using either substitution or elimination method.
5. Check the solution against the original problem statement.

Example:
The sum of two numbers is 15 and their difference is 3. Find the numbers.

Solution:
Step 1: Let the two numbers be x and y.
Step 2: Formulate the equations:
1) x + y = 15 (The sum of two numbers is 15)
2) x – y = 3 (Their difference is 3)
Step 3: Use the elimination method (add the two equations):
(x + y) + (x – y) = 15 + 3
2x = 18
Step 4: Solve for x:
x = 18 / 2
x = 9
Step 5: Substitute x = 9 into equation (1):
9 + y = 15
y = 15 – 9
y = 6
The two numbers are 9 and 6.

Word Problems on One Linear and One Quadratic Equation

These problems typically involve geometric shapes or scenarios where one relationship is linear and another involves a product or square of variables.

Steps to solve word problems on one linear and one quadratic equation:

1. Identify the unknown quantities and assign variables.
2. Translate the given conditions into one linear equation and one quadratic equation.
3. Use the substitution method to solve the system of equations.
4. Solve the resulting quadratic equation (by factorization, completing the square, or formula).
5. Substitute the obtained values back into the linear equation to find the corresponding values of the other variable.
6. Verify the solutions in the context of the problem (e.g., lengths cannot be negative).

Example:
The length of a rectangle is 3 cm more than its width. If the area of the rectangle is 40 cm², find its dimensions.

Solution:
Step 1: Let the width of the rectangle be w cm.
Step 2: The length is 3 cm more than its width, so length (l) = w + 3 cm.
Step 3: The area of a rectangle is length × width. Given area = 40 cm².
So, l × w = 40.
Step 4: Substitute l = w + 3 into the area equation:
(w + 3)w = 40
w² + 3w = 40
Step 5: Rearrange into a standard quadratic equation:
w² + 3w – 40 = 0
Step 6: Factorize the quadratic equation:
(w + 8)(w – 5) = 0
So, w = -8 or w = 5.
Step 7: Since width cannot be negative, we take w = 5 cm.
Step 8: Calculate the length:
l = w + 3 = 5 + 3 = 8 cm.
The dimensions of the rectangle are length = 8 cm and width = 5 cm.

Application to Capital Market

The capital market is a financial market where long-term funds are raised by companies and governments through the issuance of stocks and bonds. Linear and quadratic equations are used in various aspects of the capital market:

1. Supply and Demand Analysis: Linear equations can model the supply and demand curves for financial assets (e.g., stocks). The equilibrium price and quantity can be found by solving these simultaneous equations.
2. Profit and Loss Functions: Companies use quadratic equations to model profit functions, often showing how profit changes with the quantity of goods produced or services offered. Maximizing profit involves finding the vertex of a parabolic (quadratic) profit function.
3. Investment Returns: Compound interest calculations, which can lead to exponential growth, are often simplified or analyzed using quadratic approximations in certain financial models, especially when considering rates of return over time.
4. Risk Assessment: Quadratic equations are used in portfolio theory to minimize risk for a given level of return, or maximize return for a given level of risk.

Quick Revision Table

Area	Key Point
Revision of Simultaneous Linear Equations	Simultaneous linear equations are a set of two or more linear equations that involve the same variables and are solved together to find a common…
Revision of Simultaneous Linear and Quadratic Equations	This involves solving a system where one equation is linear and the other is quadratic. The primary method used is substitution.
Word Problems on Linear Equations	Word problems require careful reading to translate the given information into a single linear equation with one unknown. Steps to solve word…
Word Problems on Simultaneous Linear Equations	These problems involve two unknown quantities and relationships that can be expressed as two linear equations. Steps to solve word problems on…
Word Problems on One Linear and One Quadratic Equation	These problems typically involve geometric shapes or scenarios where one relationship is linear and another involves a product or square of…
Application to Capital Market	The capital market is a financial market where long-term funds are raised by companies and governments through the issuance of stocks and bonds….

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction and Recall

Time: 4 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets the students and reminds them of previous lessons on solving linear and quadratic equations. The teacher then introduces the topic by stating that today's lesson will focus on applying these equations to solve real-world problems.

Students’ Activity: Students respond to greetings and recall previous knowledge. They listen attentively to the introduction.

Learning Point: Prior equation knowledge.

Step 2: Revision of Simultaneous Linear Equations

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher displays charts showing examples of simultaneous linear equations and revises the substitution and elimination methods with the students, solving one example for each method on the board.

Students’ Activity: Students observe the charts, participate in the revision, and solve examples with the teacher.

Learning Point: Equation problem-solving methods.

Step 3: Revision of Simultaneous Linear and Quadratic Equations

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains how to solve simultaneous equations involving one linear and one quadratic equation, emphasizing the substitution method. An example is solved on the board.

Students’ Activity: Students pay attention to the explanation and work through the example with the teacher.

Learning Point: Equation problem-solving methods.

Step 4: Word Problems on Linear and Simultaneous Linear Equations

Time: 5 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher guides students on interpreting simple word problems into linear and simultaneous linear equations. Two examples are worked through, focusing on translating words into mathematical expressions.

Students’ Activity: Students identify keywords, help formulate equations, and contribute to solving the problems.

Learning Point: Equation problem-solving methods.

Step 5: Word Problems on One Linear and One Quadratic Equation

Time: 5 minutes

Teaching Skill: Problem-Solving Facilitation

Teacher’s Activity: The teacher presents a word problem that leads to one linear and one quadratic equation (e.g., involving area and perimeter of a rectangle). The teacher guides students through setting up and solving the equations.

Students’ Activity: Students identify the unknowns, help form the equations, and attempt to solve them collaboratively.

Learning Point: Equation problem-solving methods.

Step 6: Application to Capital Market

Time: 5 minutes

Teaching Skill: Discussion/Conceptualization

Teacher’s Activity: The teacher explains what the capital market is and discusses how linear and quadratic equations are used in areas such as supply and demand, profit analysis, and investment calculations.

Students’ Activity: Students listen, ask questions for clarification, and contribute ideas on how mathematics is applied in finance.

Learning Point: Equation practical application.

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. State two methods for solving simultaneous linear equations.
2. Explain how you would approach solving a word problem that involves one linear and one quadratic equation.
3. Give an example of how linear equations can be applied in the capital market.
4. Solve the following word problem: "The sum of two numbers is 12 and their difference is 4. Find the numbers."

Students’ Activity: Students answer orally and in writing, showing working or giving examples where the lesson requires it.

Learning Point: Equation skills assessed.

Step 8: Note-Taking

Time: 8 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides students to copy the essential Board Summary on equation into their notebooks, drawing attention to the key rules, examples, diagrams or distinctions already taught.

Students’ Activity: Students copy the essential notes carefully into their notebooks and correct any errors in their earlier work.

Learning Point: Equation notes recorded.

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Reinforcement/Summarization

Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of linear and quadratic equations in solving various real-life and financial problems. Homework is assigned.

Students’ Activity: Students listen to the summary and copy down the assigned homework.

Learning Point: Equation concepts consolidated.

Continuous Assessment/Further Study

Type: Practice Exercise

Instruction: Complete the following tasks in your notebook. Show all mathematical working clearly.

1. Form a linear equation from a suitable word problem and solve it.
2. Form and solve two simultaneous linear equations from a word problem.
3. Solve a problem involving one linear and one quadratic equation.
4. Check the solution of a quadratic equation by substitution.
5. Write a short explanation of how equations can be applied to a simple capital-market problem.

Lesson Keywords

• Linear Equation – An algebraic equation in which the highest power of the variable is 1.
• Quadratic Equation – An algebraic equation in which the highest power of the variable is 2, typically in the form ax² + bx + c = 0.
• Simultaneous Equations – A set of equations containing multiple variables that are solved together to find values for the variables that satisfy all equations.
• Substitution Method – A technique for solving simultaneous equations by expressing one variable in terms of another and replacing it in the second equation.
• Elimination Method – A technique for solving simultaneous equations by adding or subtracting equations to remove one variable.
• Word Problem – A mathematical problem presented in descriptive language rather than in mathematical notation.
• Capital Market – A financial market for long-term funds, where businesses and governments can raise money.

Differentiation

For struggling learners, the teacher will provide simpler word problems and more direct guidance on setting up equations, possibly using visual aids to represent variables. Advanced learners will be given more complex word problems or encouraged to explore additional applications of equations in finance and other fields, possibly by researching online examples.

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