Further Mathematics - Senior Secondary 1

TERM: 1^ST TERM

WEEK 1

Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes per period
Subject: Further Mathematics
Topic: Set Theory (I)
Focus: Introduction to sets, set notation, and types of sets.

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

Define what a set is.
Recognize and understand different methods of representing sets using notation.
Identify and classify various types of sets.
Understand the concept of the number of elements in a set.

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Class discussion
• Practice exercises
• Visual aids (charts)

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Charts illustrating different sets and their notations
• Flashcards with sets and types of sets
• Worksheets for set exercises

PERIOD 1 & 2: Introduction to Set Theory

PRESENTATION:

Step	Teacher’s Activity	Student’s Activity
Step 1 - Introduction to Sets	Introduces the concept of a set, explaining that a set is a well-defined collection of distinct objects. Explains that elements of a set are written in curly brackets.	Students listen attentively and ask questions for clarification.
Step 2 - Set Notation Methods	Introduces different ways to represent a set: roster (listing method) and set-builder notation. Examples: Roster: {1, 2, 3}, Set-builder: {x	x is an even number less than 5}.
Step 3 - Types of Sets	Explains the types of sets: a) Null set (empty set) – A set with no elements. b) Singleton set – A set with only one element. c) Finite set – A set with a countable number of elements. d) Infinite set – A set with an uncountable number of elements. e) Universal set – A set containing all elements under consideration for a particular context. f) Power set – The set of all subsets of a given set. g) Number of elements in a set – Cardinality of the set.	Students observe and write down the definitions and examples for each type of set.
Step 4 - Analogies and Real-Life Connections	Uses analogies to relate sets to real-life examples: e.g., a set of all students in the class (finite set), the set of all natural numbers (infinite set).	Students share their own analogies and examples related to sets.

NOTE ON BOARD:

Definition of a Set: A set is a well-defined collection of distinct objects.
Set Notations:

Roster Method: {a, b, c}
Set-builder Notation: {x | x is a prime number less than 10}

Types of Sets:

Null Set: Ø or {}
Singleton Set: {1}
Finite Set: {1, 2, 3}
Infinite Set: {1, 2, 3, ...}
Universal Set: U = {all elements under consideration}
Power Set: P(A) = {Ø, {1}, {2}, {1, 2}}

Number of Elements: The number of elements in a set is called its cardinality, denoted |A|.

Students copy the notes from the board.

EVALUATION (5 exercises):

Define a set and provide an example.
What does the cardinality of a set represent?
What is the difference between a finite and an infinite set?
Give an example of a universal set in your context.
Explain the power set of the set {a, b}.

CLASSWORK (5 questions):

Write down the set of even numbers between 1 and 10 in roster form.
Identify if the set {1, 2, 3} is finite or infinite.
Write the set of all prime numbers less than 20 in set-builder notation.
How many elements are in the set {apple, banana, mango}?
List the power set of {1, 2}.

ASSIGNMENT (5 tasks):

Research and define a null set with an example.
Explain why {a} is a singleton set.
Write the universal set for the days of the week.
Find the power set of {a, b, c}.
What is the cardinality of the set of all odd numbers between 1 and 100?

PERIOD 3 & 4: Advanced Set Operations and Notation

PRESENTATION:

Step	Teacher’s Activity	Student’s Activity
Step 1 - Recap of Set Types	Quickly reviews the different types of sets from the previous lesson, focusing on any questions or clarifications needed.	Students engage with the teacher for clarification.
Step 2 - Operations on Sets	Introduces set operations: union (A ∪ B), intersection (A ∩ B), and difference (A - B). Uses visual aids (Venn diagrams) to explain these operations.	Students follow the teacher's demonstration, take notes, and participate in class exercises.
Step 3 - Number of Elements in Power Set	Explains how to calculate the number of elements in a power set using the formula	P(A)
Step 4 - Guided Practice	Provides examples of set operations (union, intersection, difference) and asks students to solve in pairs.	Students practice solving set operation problems in pairs.

NOTE ON BOARD:

Union of Sets (A ∪ B): The set containing all elements from both sets A and B.
Intersection of Sets (A ∩ B): The set containing only the elements that are in both sets A and B.
Difference of Sets (A - B): The set containing elements that are in set A but not in set B.
Power Set Formula: If A has n elements, then the number of elements in the power set of A is 2ⁿ.

EVALUATION (5 exercises):

What is the union of sets {1, 2, 3} and {3, 4, 5}?
What is the intersection of sets {a, b, c} and {b, c, d}?
Find the difference between the sets {1, 2, 3, 4} and {2, 3}.
How many elements are in the power set of {a, b}?
Write the union of {2, 4, 6} and {1, 3, 5}.

CLASSWORK (5 questions):

Find the intersection of {1, 2, 3} and {3, 4, 5}.
Find the difference between {a, b, c} and {b, c, d}.
Find the union of {x, y} and {z}.
How many elements are in the power set of {a, b, c, d}?
List the elements of the power set of {1, 2}.

ASSIGNMENT (5 tasks):

Find the union of {1, 2, 3} and {2, 3, 4}.
Find the intersection of {a, b} and {b, c}.
List the power set of {p, q}.
If A = {1, 2, 3, 4}, find the number of elements in the power set of A.

Solve for A ∩ B when A = {1, 3, 5} and B = {3, 4, 5}.

Further Mathematics - Senior Secondary 1 - Set theory (I)