Lesson Note on Quadratic Equation: Word Problems and Real-Life Applications for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 1st Term
Week: 9
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: Quadratic Equation: Completing the Square and Formula.
Topic: QUADRATIC EQUATION
Subject Matter: Word problem leading to quadratic equation, application of quadratic equation to real life situation

Specific Objectives

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

Cognitive Domain:

• State the formula for forming a quadratic equation given its roots.
• Transform word problems into quadratic equations.
• Solve quadratic equations formed from real-life situations.

Affective Domain:

• Appreciate the importance of quadratic equations in solving real-world problems.
• Develop interest in applying mathematical concepts to practical scenarios.

Psychomotor Domain:

• Accurately form quadratic equations from given roots or word problems.
• Demonstrate the steps for solving quadratic equations derived from word problems.

Social Domain:

• Collaborate with peers to discuss and solve word problems.
• Communicate mathematical ideas clearly when explaining solutions.

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:

• Whiteboard/Blackboard
• Markers/Chalk
• Calculators
• Examples of financial reports and budget data
• Population figures and data from school records

Rationale for the Lesson

This lesson helps pupils understand how to translate everyday problems into mathematical equations. It enables them to apply quadratic equations to solve practical situations encountered in various fields such as finance, science, and engineering.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of solving linear equations, basic algebraic expressions, and solving quadratic equations using various methods (factorization, completing the square, general formula).

Lesson Content/Board Summary

QUADRATIC EQUATION

1. Formation of Quadratic Equations from Sum and Product of Roots

A quadratic equation can be formed if the sum (S) and product (P) of its roots are known. The general form of such an equation is:

x² – (sum of roots)x + (product of roots) = 0

Or, x² – Sx + P = 0

If α and β are the roots of a quadratic equation ax² + bx + c = 0, then:

• Sum of roots (S) = α + β = -b/a
• Product of roots (P) = αβ = c/a

Worked Examples:

Example 1: Form a quadratic equation whose roots are 3 and -5.

Solution:

Step 1: Identify the roots. Let α = 3 and β = -5.

Step 2: Calculate the sum of roots (S).

S = α + β = 3 + (-5) = 3 – 5 = -2

Step 3: Calculate the product of roots (P).

P = αβ = 3 × (-5) = -15

Step 4: Substitute S and P into the formula x² – Sx + P = 0.

x² – (-2)x + (-15) = 0

x² + 2x – 15 = 0

Therefore, the quadratic equation is x² + 2x – 15 = 0.

Example 2: If the roots of a quadratic equation are 2/3 and -1/2, find the equation.

Solution:

Step 1: Identify the roots. Let α = 2/3 and β = -1/2.

Step 2: Calculate the sum of roots (S).

S = α + β = 2/3 + (-1/2) = 2/3 – 1/2 = (4 – 3)/6 = 1/6

Step 3: Calculate the product of roots (P).

P = αβ = (2/3) × (-1/2) = -2/6 = -1/3

Step 4: Substitute S and P into the formula x² – Sx + P = 0.

x² – (1/6)x + (-1/3) = 0

x² – (1/6)x – 1/3 = 0

Step 5: Multiply through by the LCM (6) to clear fractions.

6(x²) – 6(1/6)x – 6(1/3) = 0

6x² – x – 2 = 0

Therefore, the quadratic equation is 6x² – x – 2 = 0.

2. Transforming Word Problems into Quadratic Equations

Many real-life situations can be modeled and solved using quadratic equations. The key is to correctly translate the verbal description into an algebraic equation.

Steps to follow:

• Read the problem carefully to understand the situation.
• Identify the unknown quantity and assign a variable (e.g., x) to it.
• Translate the relationships described in the problem into algebraic expressions.
• Formulate a quadratic equation based on these expressions.
• Solve the quadratic equation using an appropriate method (factorization, formula, completing the square).
• Check if the solution makes sense in the context of the original word problem (e.g., length cannot be negative).

Worked Examples:

Example 1: The product of two consecutive positive integers is 72. Find the integers.

Solution:

Step 1: Define the unknown integers. Let the first positive integer be x.

Step 2: The next consecutive positive integer will be x + 1.

Step 3: Formulate the equation based on the given information (product is 72).

x(x + 1) = 72

Step 4: Expand and rearrange into the standard quadratic form ax² + bx + c = 0.

x² + x = 72

x² + x – 72 = 0

Step 5: Solve the quadratic equation (by factorization).

(x + 9)(x – 8) = 0

So, x + 9 = 0 or x – 8 = 0

x = -9 or x = 8

Step 6: Check the solution. Since the integers must be positive, x = 8 is the valid solution.

The first integer is 8.

The second integer is x + 1 = 8 + 1 = 9.

Check: 8 × 9 = 72. (Correct)

Therefore, the two consecutive positive integers are 8 and 9.

Example 2: A rectangular garden has a length that is 5 meters more than its width. If the area of the garden is 84 square meters, find the dimensions of the garden.

Solution:

Step 1: Define the unknown dimensions. Let the width of the garden be w meters.

Step 2: The length of the garden is 5 meters more than its width, so length = w + 5 meters.

Step 3: Formulate the equation using the area formula (Area = length × width).

w(w + 5) = 84

Step 4: Expand and rearrange into the standard quadratic form.

w² + 5w = 84

w² + 5w – 84 = 0

Step 5: Solve the quadratic equation (by factorization).

(w + 12)(w – 7) = 0

So, w + 12 = 0 or w – 7 = 0

w = -12 or w = 7

Step 6: Check the solution. Since width cannot be negative, w = 7 meters is the valid solution.

The width of the garden is 7 meters.

The length of the garden is w + 5 = 7 + 5 = 12 meters.

Check: Area = 12 × 7 = 84 sq meters. (Correct)

Therefore, the dimensions of the garden are 7 meters by 12 meters.

3. Applications of Quadratic Equations to Real-Life Situations

Quadratic equations are used in various fields to model and solve problems. Some common applications include:

• Physics: Projectile motion, calculating the path of objects in flight.
• Engineering: Designing structures, optimizing processes.
• Economics and Finance: Calculating profit, loss, and pricing strategies.
• Area and Geometry: Finding dimensions of shapes.
• Optimization Problems: Finding maximum or minimum values in various scenarios.

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 reminds them of their previous lesson on solving quadratic equations. The teacher then asks pupils to recall the general form of a quadratic equation.
Pupils’ Activity: Pupils respond to the greetings and state the general form of a quadratic equation (ax² + bx + c = 0).
Learning Point: Pupils recall prior knowledge of quadratic equations.

Step 2: Review of Quadratic Equation Basics

Time: 5 minutes
Teaching Skill: Recall/Review
Teacher’s Activity: The teacher briefly reviews the methods of solving quadratic equations (factorization, formula, completing the square) and emphasizes their importance.
Pupils’ Activity: Pupils listen attentively and contribute by mentioning the methods they remember.
Learning Point: Pupils refresh their understanding of solving quadratic equations.

Step 3: Formation of Quadratic Equations from Roots

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the formula x² – Sx + P = 0 for forming a quadratic equation from its roots. The teacher demonstrates with Example 1 from the board summary, explaining each step clearly.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils learn how to form quadratic equations given the sum and product of their roots.

Step 4: Pupils Practice Formation

Time: 7 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides another example (e.g., roots are -4 and 6) and guides pupils to work through it on their own or in pairs.
Pupils’ Activity: Pupils attempt to form the quadratic equation for the given roots, applying the learned formula.
Learning Point: Pupils practice and consolidate their understanding of forming quadratic equations from roots.

Step 5: Transforming Word Problems into Quadratic Equations

Time: 8 minutes
Teaching Skill: Explanation/Problem Solving
Teacher’s Activity: The teacher explains the steps involved in transforming word problems into quadratic equations. The teacher then works through Example 1 from the board summary, showing how to define variables, form the equation, and solve it, emphasizing the interpretation of the solution.
Pupils’ Activity: Pupils pay close attention to the steps and the teacher’s problem-solving approach.
Learning Point: Pupils learn the process of converting word problems into solvable quadratic equations.

Step 6: Pupils Practice Word Problems

Time: 7 minutes
Teaching Skill: Independent Practice
Teacher’s Activity: The teacher presents another word problem (e.g., Example 2 from the board summary) and asks pupils to attempt it, offering guidance where needed.
Pupils’ Activity: Pupils work individually or in small groups to set up and solve the quadratic equation from the word problem.
Learning Point: Pupils apply their skills to solve word problems involving quadratic equations.

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 the formula for forming a quadratic equation given its roots.
2. Form a quadratic equation whose roots are 5 and -2.
3. The sum of a number and its square is 42. Form a quadratic equation to represent this statement.
4. A rectangle has a length of (x + 3) cm and a width of x cm. If its area is 40 cm², write down the quadratic equation that describes this situation.

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, reiterating how to form quadratic equations from roots and how to translate word problems into quadratic equations. The teacher assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils reinforce their understanding of the lesson and are prepared for further practice.

Lesson Keywords

• Quadratic Equation – An equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Roots – The values of the variable that satisfy a quadratic equation; also known as solutions or zeros.
• Sum of Roots – For ax² + bx + c = 0, the sum of roots is -b/a.
• Product of Roots – For ax² + bx + c = 0, the product of roots is c/a.
• Word Problem – A mathematical problem presented in descriptive language, requiring translation into an equation.

Differentiation

For struggling learners, the teacher will provide simpler word problems with direct translations and more guided support during problem-solving. Advanced learners will be challenged with more complex word problems or asked to explore applications of quadratic equations in other subjects like Physics or Economics.

Note for teachers using this lesson plan

Ensure that pupils have a strong grasp of basic quadratic equation solving methods before introducing word problems. Encourage pupils to draw diagrams for word problems involving dimensions or physical layouts. Emphasize checking solutions within the context of the word problem to discard unreasonable answers (e.g., negative lengths or ages).