Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 3rd Term
Week: 6
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Models II.
Topic: MODELS I
Subject Matter: models of operations research, linear programming models, transportation models, assignment models

Specific Objectives

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

Cognitive Domain:

Define Operations Research (OR) models.
Identify different types of Operations Research models.
Explain the characteristics of linear programming, transportation, and assignment models.

Affective Domain:

Appreciate the importance of Operations Research models in solving real-world problems.
Develop an interest in applying mathematical models for decision-making.

Psychomotor Domain:

Formulate simple linear programming models from given scenarios.
Construct basic transportation and assignment model structures.

Collaborate with peers to discuss and analyze different model types.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Further Mathematics (Senior Secondary Education).
State Unified Scheme of Work for Further Mathematics.
New Further Mathematics Project 1 by P.N. Okafor et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts illustrating operations research models.
Whiteboard and markers/chalk.
Further Mathematics textbooks.

Rationale for the Lesson

This lesson helps pupils understand how mathematical models are used to solve practical problems in various fields. It enables them to see the real-life application of mathematical concepts, which can improve their decision-making skills in different situations.

Prerequisite/Previous Knowledge

Pupils are expected to have a basic understanding of mathematical concepts such as equations, inequalities, and problem-solving techniques.

Lesson Content/Board Summary

MODELS I

Introduction to Operations Research (OR) Models

Operations Research (OR) involves the application of scientific methods and techniques to complex problems concerning the management of large systems of people, machines, materials, and money in industry, business, government, and defense. The objective is to provide quantitative bases for decision-making.

Types of Operations Research Models

Operations Research models are broadly classified based on their structure and the nature of the problem they address. The following are common types of OR models:

Linear Programming Models
Transportation Models
Assignment Models

Linear Programming Models

Linear Programming (LP) is a mathematical technique for determining an optimal allocation of resources in a situation where there are alternative courses of action. It is used to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

The key components of an LP model are:

Objective Function: A linear mathematical expression that describes the problem’s goal (e.g., maximize profit, minimize cost).
Decision Variables: The unknown quantities that the model seeks to determine.
Constraints: Linear inequalities or equalities that limit the available resources or represent requirements.
Non-negativity Restrictions: Decision variables must be non-negative.

Transportation Models

Transportation models are a special class of linear programming problems designed to minimize the cost of distributing a product from a number of sources (factories) to a number of destinations (warehouses or retail outlets). The model assumes that the cost of transportation is directly proportional to the number of units transported.

The key elements of a transportation model are:

Sources: Locations from which goods are shipped (e.g., factories, plants) with specified supply capacities.
Destinations: Locations to which goods are shipped (e.g., warehouses, markets) with specified demand requirements.
Unit Transportation Costs: The cost of shipping one unit of the product from each source to each destination.

Assignment Models

Assignment models are a special case of transportation models where the objective is to assign a number of tasks (jobs) to an equal number of agents (workers, machines) such that the total cost (or time) is minimized, or total profit (or efficiency) is maximized. Each agent is assigned to exactly one task, and each task is assigned to exactly one agent.

The key elements of an assignment model are:

Tasks: The jobs or activities that need to be performed.
Agents: The resources available to perform the tasks (e.g., workers, machines).
Cost/Profit Matrix: A matrix showing the cost or profit associated with assigning each agent to each task.

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 asks them to recall situations where they had to choose the best option from several alternatives (e.g., choosing the shortest route to school, allocating chores at home). The teacher then introduces the topic “MODELS I” and explains that mathematical models can help make such decisions scientifically.
Pupils’ Activity: Pupils respond to the teacher’s questions, share their experiences, and listen attentively to the introduction.
Learning Point: Pupils connect the lesson to real-life decision-making scenarios.

Step 2: Introduction to Operations Research (OR) Models

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines Operations Research (OR) and explains the general purpose of OR models. The teacher also introduces the concept of using models to represent real-world problems for analysis.
Pupils’ Activity: Pupils listen, ask questions for clarification, and take notes on the definition and purpose of OR models.
Learning Point: Pupils understand what Operations Research models are and why they are used.

Step 3: Linear Programming Models

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains Linear Programming (LP) models, outlining their definition and key components (objective function, decision variables, constraints, non-negativity). The teacher provides a simple hypothetical scenario and guides pupils to identify these components to formulate a basic LP model.
Pupils’ Activity: Pupils listen, identify components in the given scenario, and attempt to formulate simple LP models under the teacher’s guidance.
Learning Point: Pupils can define LP models, identify their components, and formulate simple LP problems.

Step 4: Transportation Models

Time: 7 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains transportation models, highlighting their purpose (minimizing distribution costs) and key elements (sources, destinations, supply, demand, unit costs). The teacher uses a chart or draws a simple diagram to illustrate a transportation problem.
Pupils’ Activity: Pupils listen, observe the illustrations, and identify the key elements of a transportation problem in given scenarios.
Learning Point: Pupils understand the concept and elements of transportation models.

Step 5: Assignment Models

Time: 7 minutes
Teaching Skill: Explanation/Comparison
Teacher’s Activity: The teacher explains assignment models, their purpose (optimal allocation of tasks to agents), and key elements (tasks, agents, cost/profit matrix). The teacher also highlights similarities and differences between assignment and transportation models.
Pupils’ Activity: Pupils listen, ask questions, and identify the key elements of an assignment problem.
Learning Point: Pupils understand the concept and elements of assignment models.

Step 6: Class Activity/Practice

Time: 4 minutes
Teaching Skill: Application
Teacher’s Activity: The teacher presents a short problem scenario and asks pupils to identify which type of OR model (linear programming, transportation, or assignment) would be most suitable to solve it, and to list the key elements involved.
Pupils’ Activity: Pupils discuss in pairs or small groups and identify the model type and elements.
Learning Point: Pupils apply their knowledge to classify problem types and identify relevant elements.

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 (OR) models.
Mention two types of Operations Research models.
State the key components of a Linear Programming model.
Explain the main purpose of a Transportation model.

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 main points of the lesson, emphasizing the different types of OR models discussed and their practical applications. The teacher assigns homework, which could involve finding real-life examples of each model type.
Pupils’ Activity: Pupils listen to the summary and note down the homework.
Learning Point: Reinforces the main concepts of the lesson.

Lesson Keywords

Operations Research (OR) – A scientific approach to decision-making that seeks to determine how to design and operate a system most effectively.
Models – Simplified representations of real-world systems or problems.
Linear Programming – A mathematical method for determining an optimal plan of action in problems where the objective function and constraints are linear.
Transportation Model – An OR model used to minimize the cost of shipping goods from sources to destinations.
Assignment Model – An OR model used to assign tasks to agents to minimize cost or maximize profit, with each task assigned to one agent.
Objective Function – The mathematical expression that defines the goal of an optimization problem (e.g., maximize profit, minimize cost).
Constraints – Limitations or restrictions that must be satisfied in an optimization problem.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex scenarios for model formulation. For those who require more support, the teacher can offer simplified examples, provide guided practice, or pair them with stronger pupils for peer tutoring.

Note for teachers using this lesson plan

Teachers should ensure to use practical, relatable examples from the Nigerian context to illustrate the various models. Visual aids like charts and diagrams are important for clarifying abstract concepts. Encourage pupils to ask questions and engage in discussions to deepen their understanding.