Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: Second Term
Week: 3
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Functions II.
Topic: FUNCTIONS
Subject Matter: Definition of a function, one-to-one function, onto function, inverse function, identity function, constant function, circular function.

Specific Objectives

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

Cognitive Domain:

Define a function.
Identify different types of functions.
Distinguish between one-to-one and onto functions.
Explain the concept of an inverse function.

Affective Domain:

Appreciate the importance of functions in mathematics.
Show interest in solving problems involving functions.
Collaborate with peers to discuss examples of functions.

Psychomotor Domain:

Sketch simple graphical representations of functions.
Determine if a given relation is a function.
Calculate the inverse of a simple function.

Participate actively in class discussions.
Communicate mathematical ideas clearly to others.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Further Mathematics.
State Unified Scheme of Work for Senior Secondary Schools.
New Further Mathematics for Senior Secondary Schools by P.N. Okeke.
Comprehensive Further Mathematics for Senior Secondary Schools by A.B. Oyedele.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts illustrating different types of functions.
Whiteboard and markers/chalk.
Relevant mathematical textbooks.

Rationale for the Lesson

Understanding functions is important for pupils because functions describe relationships between quantities, which is a fundamental concept in mathematics and science. This knowledge helps pupils to solve problems in various fields and prepares them for more advanced mathematical topics.

Prerequisite/Previous Knowledge

Pupils are expected to have a basic understanding of sets, relations, and mapping from their core mathematics lessons.

Lesson Content/Board Summary

FUNCTIONS

Definition of a Function

A function is a special type of relation where each element in the domain (input) is mapped to exactly one element in the codomain (output). It can be represented as f: X → Y, where X is the domain and Y is the codomain.

One-to-One Function (Injective Function)

A function f: X → Y is a one-to-one function if distinct elements in the domain X are mapped to distinct elements in the codomain Y. This means that if f(x₁) = f(x₂), then x₁ = x₂.

Onto Function (Surjective Function)

A function f: X → Y is an onto function if every element in the codomain Y is the image of at least one element in the domain X. This means that the range of the function is equal to its codomain.

Inverse Function

If a function f: X → Y is both one-to-one and onto (bijective), then an inverse function, denoted as f⁻¹: Y → X, exists. The inverse function reverses the mapping of the original function such that f⁻¹(y) = x if and only if f(x) = y.

Identity Function

An identity function, denoted as I(x) or id(x), maps every element to itself. For any element x in the domain, I(x) = x. The graph of an identity function is a straight line passing through the origin with a slope of 1 (y=x).

Constant Function

A constant function is a function whose output value is the same for every input value. If f(x) = c, where c is a fixed real number, then f is a constant function. The graph of a constant function is a horizontal line.

Circular Functions (Trigonometric Functions)

Circular functions are functions of an angle that relate the angles of a triangle to the lengths of its sides. They are periodic functions related to the unit circle. The following are common circular functions:

Sine (sin x)
Cosine (cos x)
Tangent (tan x)
Cosecant (csc x)
Secant (sec x)
Cotangent (cot x)

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 reviews the concept of relations, asking pupils to give examples. The teacher then introduces the topic of functions as a special type of relation.
Pupils’ Activity: Pupils respond to greetings, recall previous knowledge about relations, and listen attentively to the introduction.
Learning Point: Pupils recall prior knowledge of relations and are introduced to the concept of functions.

Step 2: Definition of a Function

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines a function, explaining that each input has exactly one output. The teacher uses diagrams to illustrate the difference between a relation that is a function and one that is not.
Pupils’ Activity: Pupils listen, take notes, and ask clarifying questions. They attempt to define a function in their own words.
Learning Point: Pupils understand the formal definition of a function and can identify functions from diagrams.

Step 3: One-to-One and Onto Functions

Time: 8 minutes
Teaching Skill: Elaboration/Classification
Teacher’s Activity: The teacher explains one-to-one functions and onto functions with clear examples and diagrams. The teacher emphasizes the conditions for each type of function.
Pupils’ Activity: Pupils observe the diagrams, listen to explanations, and identify examples of one-to-one and onto functions.
Learning Point: Pupils can differentiate between one-to-one and onto functions.

Step 4: Inverse and Identity Functions

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines an inverse function and explains the condition for its existence (bijective function). The teacher also defines and gives examples of an identity function.
Pupils’ Activity: Pupils listen, take notes, and ask questions about the conditions for inverse functions. They identify the identity function.
Learning Point: Pupils understand what inverse and identity functions are and when an inverse function exists.

Step 5: Constant and Circular Functions

Time: 8 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher defines a constant function and illustrates its graph. The teacher then introduces circular (trigonometric) functions, listing common examples and their basic properties.
Pupils’ Activity: Pupils listen, observe the constant function graph, and note down the examples of circular functions.
Learning Point: Pupils can define constant functions and list common circular functions.

Step 6: Application and Practice

Time: 5 minutes
Teaching Skill: Problem Solving/Guided Practice
Teacher’s Activity: The teacher provides a few examples on the board, asking pupils to identify the type of function or to determine if a given relation is a function. The teacher guides them through the solution process.
Pupils’ Activity: Pupils attempt to classify functions and solve problems based on the definitions learned. They ask for clarification where needed.
Learning Point: Pupils apply their understanding to identify and classify functions.

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 a function.
State two types of functions.
Explain the difference between a one-to-one function and an onto function.
When does an inverse function exist?

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, reinforcing the definitions and types of functions discussed. The teacher assigns homework involving identifying and classifying functions.
Pupils’ Activity: Pupils listen to the summary and copy the assigned homework.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

Function – A relation where each input has exactly one output.
One-to-one function – Each distinct input maps to a distinct output.
Onto function – Every element in the codomain is an output of at least one input.
Inverse function – A function that reverses the effect of the original function.
Identity function – A function that maps every element to itself, f(x) = x.
Constant function – A function whose output value is always the same, f(x) = c.
Circular function – Functions of an angle related to the unit circle, e.g., sine, cosine.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex examples or challenge them to find the inverse of more involved functions. For pupils who need more support, the teacher will use simpler examples, provide additional visual aids, and offer one-on-one guidance or peer tutoring opportunities.

Note for teachers using this lesson plan

Ensure that pupils have a solid grasp of basic set theory and relations before delving into functions. Use clear, simple diagrams to illustrate the concepts, especially for one-to-one, onto, and inverse functions. Encourage pupils to provide their own real-life examples where possible to make the learning more relatable.