Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: Third Term
Week: 7
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Binary Operations I.
Topic: MODELS II
Subject Matter: Practical application of operations research models, solving real life model problems

Specific Objectives

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

Cognitive Domain:

Define operations research models.
State the general steps involved in solving practical model problems.
Identify the components of a simple linear programming model.

Affective Domain:

Appreciate the importance of operations research models in solving real-life problems.
Show willingness to apply mathematical concepts to practical situations.

Psychomotor Domain:

Formulate simple mathematical models from given real-life scenarios.
Solve basic operations research model problems.

Collaborate with peers to discuss and solve model problems.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Education Curriculum
State Unified Scheme of Work
New General Mathematics for Senior Secondary Schools, Book 1

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts of solved model problems
Whiteboard and markers
Textbooks

Rationale for the Lesson

This lesson helps pupils understand how mathematical models can be used to solve everyday problems and make better decisions. It enables them to see the practical relevance of mathematics beyond theoretical concepts, preparing them for real-world applications.

Prerequisite/Previous Knowledge

Pupils should have basic knowledge of algebraic equations, inequalities, and graphical representation of linear equations.

Lesson Content/Board Summary

OPERATIONS RESEARCH MODELS AND THEIR PRACTICAL APPLICATIONS

Introduction to Operations Research Models

Operations Research (OR) is a scientific approach to decision-making that involves developing mathematical models to represent complex real-world problems. These models help in finding optimal or near-optimal solutions.

Types of Operations Research Models (Linear Programming)

Linear Programming (LP) is a widely used operations research technique for optimizing a linear objective function, subject to linear equality and inequality constraints. It is used to allocate limited resources among competing activities.

The main components of a Linear Programming Model are:

Objective Function: A mathematical expression that defines the goal (e.g., maximize profit, minimize cost).
Decision Variables: The quantities that need to be determined (e.g., number of units to produce).
Constraints: Limitations or restrictions on the resources or activities (e.g., available labor, raw materials).

General Steps in Solving Practical Model Problems

The following are the general steps:

Problem Definition: Clearly understand the problem and identify the objective.
Model Formulation: Translate the real-life problem into a mathematical model (objective function, decision variables, constraints).
Solution Method: Apply an appropriate mathematical technique to solve the model (e.g., graphical method, simplex method).
Interpretation of Results: Translate the mathematical solution back into the context of the real-life problem.
Implementation: Put the solution into practice.

Practical Application of Operations Research Models

Operations Research models can be applied in various fields such as production planning, resource allocation, transportation, scheduling, and finance.

Example: A company produces two types of pens, A and B. Pen A requires 2 hours of assembly and 1 hour of finishing. Pen B requires 3 hours of assembly and 1 hour of finishing. The company has 120 hours available for assembly and 60 hours for finishing per week. If the profit from Pen A is N50 and Pen B is N70, how many of each pen should be produced to maximize profit?

Model Formulation:

Let x = number of Pen A, y = number of Pen B.
Objective Function: Maximize P = 50x + 70y (Profit)
Constraints:
- 2x + 3y ≤ 120 (Assembly hours)
- 1x + 1y ≤ 60 (Finishing hours)
- x ≥ 0, y ≥ 0 (Non-negativity)

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 introduces the topic “MODELS II” by asking pupils to recall what a model is in general terms and how mathematical models can represent real-life situations. The teacher explains that this lesson will focus on practical applications of such models.
Pupils’ Activity: Pupils respond to questions and listen attentively.
Learning Point: Pupils connect the concept of models to real-life problem-solving.

Step 2: Explanation of Operations Research Models

Time: 8 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines Operations Research (OR) and explains its purpose. The teacher then introduces Linear Programming as a key type of OR model, detailing its objective function, decision variables, and constraints, using the board.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and basic components of operations research and linear programming models.

Step 3: Steps in Model Formulation and Solution

Time: 8 minutes
Teaching Skill: Elucidation/Illustration
Teacher’s Activity: The teacher outlines and explains the general steps involved in solving practical model problems, emphasizing how a real-world problem is translated into a mathematical model. Charts of solved model problems are used as visual aids.
Pupils’ Activity: Pupils observe the charts, listen to the explanation, and copy the steps.
Learning Point: Pupils learn the systematic approach to solving model problems.

Step 4: Practical Application Example 1 (Teacher-led)

Time: 10 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher presents a simple real-life problem (e.g., the pen production example from the board summary) and guides pupils step-by-step through its formulation into a linear programming model. The teacher shows how to identify the objective function, decision variables, and constraints.
Pupils’ Activity: Pupils follow along, ask questions, and attempt to formulate the model as guided by the teacher.
Learning Point: Pupils practice formulating mathematical models from practical scenarios.

Step 5: Practical Application Example 2 (Pupil Participation)

Time: 8 minutes
Teaching Skill: Collaborative Learning
Teacher’s Activity: The teacher provides another similar real-life problem and asks pupils to work in pairs or small groups to formulate the mathematical model. The teacher moves around, providing assistance and feedback.
Pupils’ Activity: Pupils discuss and collaborate to formulate the model for the given problem.
Learning Point: Pupils apply their understanding to solve a new problem collaboratively.

Step 6: Review of Pupil Solutions

Time: 3 minutes
Teaching Skill: Feedback/Correction
Teacher’s Activity: The teacher calls on a few groups to present their formulated models. The teacher reviews and corrects any errors or misunderstandings on the board, reinforcing the correct approach.
Pupils’ Activity: Pupils present their work, listen to feedback, and make corrections.
Learning Point: Pupils receive immediate feedback and clarify any misconceptions.

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 Operations Research Models.
Mention any three general steps involved in solving practical model problems.
State the three main components of a Linear Programming Model.
Give an example of a real-life situation where operations research models can be applied.

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 learning points of the lesson, reiterating the importance of operations research models in practical decision-making. The teacher assigns homework, which may include finding and formulating another simple real-life problem.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their learning and receive further practice.

Lesson Keywords

Operations Research – A scientific approach to decision-making using mathematical models.
Model – A mathematical representation of a real-world problem.
Linear Programming – An optimization technique for linear objective functions and constraints.
Objective Function – The mathematical expression representing the goal (e.g., profit, cost).
Decision Variables – The unknown quantities to be determined in a model.
Constraints – Limitations or restrictions in a model.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex model problems or encourage them to explore different solution methods. For pupils needing more support, the teacher can provide pre-formulated parts of models or simpler, guided examples, offering one-on-one assistance during group activities.

Note for teachers using this lesson plan

Ensure that the real-life examples chosen are relatable and easy for pupils to understand. Emphasize the systematic approach to problem-solving. Encourage active participation and discussion, especially during model formulation, to build confidence in applying mathematical concepts.