Mathematics - Senior Secondary 2 - Logical Reasoning (Simple and Compound Statements, Logical Operations, Truth Tables, Conditional Statements, Indirect Proofs)

Logical Reasoning (Simple and Compound Statements, Logical Operations, Truth Tables, Conditional Statements, Indirect Proofs)

Get it on Google Play
Get it on the Apple App Store
  1. Home
  2. Comprehensive Lesson Plan and Notes
  3. Senior Secondary 2
  4. Logical Reasoning (Simple and Compound Statements, Logical Operations, Truth Tables, Conditional Statements, Indirect Proofs)

TERM: 2ND TERM

Week: 1

Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes per period (5 periods)
Subject: Mathematics
Topic: Logical Reasoning (Simple and Compound Statements, Logical Operations, Truth Tables, Conditional Statements, Indirect Proofs)

SPECIFIC OBJECTIVES:

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

  1. Identify and understand simple and compound statements.
  2. Perform logical operations and construct truth tables for logical statements.
  3. Recognize and apply the converse, inverse, and contrapositive of conditional statements.
  4. Understand and apply indirect proofs in logic.

 

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practice exercises
  • Analogies and real-life connections

 

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Truth table charts
  • Worksheets for truth table construction
  • Examples of logical operations and statements

 

PERIOD 1 & 2: Simple and Compound Statements

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Introduction

Introduces the concept of simple and compound statements. Simple statements involve basic assertions, while compound statements connect two or more simple statements with logical connectors (e.g., "and," "or").

Students listen attentively and ask clarifying questions.

Step 2: Examples

Provides examples of simple and compound statements. For instance: Simple: "The sky is blue." Compound: "It is raining and the sky is cloudy."

Students observe examples and participate in identifying simple and compound statements.

Step 3: Explanation

Explains the logical connectors used in compound statements, such as "and," "or," and "not."

Students take notes and discuss other examples provided.

Step 4: Guided Practice

Encourages students to create their own simple and compound statements using real-life scenarios.

Students write their own examples of simple and compound statements.

NOTE ON BOARD:

  • Simple Statements: e.g., "The sky is blue."
  • Compound Statements: e.g., "It is raining and the sky is cloudy."

EVALUATION (5 exercises):

  1. What is a simple statement? Give an example.
  2. What is a compound statement? Give an example.
  3. Identify the logical connector in the statement: "It is sunny or cloudy."
  4. Give an example of a compound statement using "and."
  5. Identify the simple statement in the compound statement: "The car is red and the tires are black."

CLASSWORK (5 questions):

  1. Identify whether the statement "I like pizza and I like pasta" is simple or compound.
  2. Create a compound statement using "not."
  3. What logical connector is used in the statement "The door is open or closed"?
  4. Create a compound statement using "and."
  5. What is the logical connector used in the statement "He is tall and strong"?

ASSIGNMENT (5 tasks):

  1. Write a simple statement about your school.
  2. Create a compound statement about the weather.
  3. Identify a logical connector in the statement: "She is smart but lazy."
  4. Write a compound statement using "or."
  5. Explain why the statement "It is raining and windy" is a compound statement.

 

PERIOD 3 & 4: Logical Operations and Truth Tables

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Introduction

Introduces logical operations such as conjunction ("and"), disjunction ("or"), negation ("not"), implication ("if...then"), and biconditional ("if and only if").

Students listen and take notes on logical operations.

Step 2: Truth Tables

Demonstrates how to construct truth tables for each logical operation. For example, the truth table for conjunction (AND) with two statements: P and Q.

Students observe the truth table construction and ask questions.

Step 3: Guided Practice

Constructs truth tables for various logical operations on the board and works through examples.

Students participate by helping fill out truth tables with the teacher.

Step 4: Independent Practice

Assigns a set of logical operations for students to complete in pairs.

Students work in pairs to complete the truth tables.

NOTE ON BOARD:

  • Conjunction (AND): P ∧ Q (True when both P and Q are true)
  • Disjunction (OR): P ∨ Q (True when either P or Q is true)
  • Negation (NOT): ¬P (True when P is false)
  • Implication (IF...THEN): P → Q (True unless P is true and Q is false)
  • Biconditional (IF AND ONLY IF): P ↔ Q (True when both P and Q have the same truth value)

EVALUATION (5 exercises):

  1. Complete the truth table for P ∧
  2. Complete the truth table for P ∨
  3. Complete the truth table for ¬P.
  4. Construct the truth table for P → Q.
  5. Construct the truth table for P ↔ Q.

CLASSWORK (5 questions):

  1. Construct a truth table for P ∧ Q with P: True, False and Q: True, False.
  2. Construct a truth table for P ∨ Q with P: True, False and Q: True, False.
  3. What is the truth value of P → Q if P is true and Q is false?
  4. What is the truth value of P ↔ Q if both P and Q are true?
  5. Construct a truth table for ¬P when P is False.

ASSIGNMENT (5 tasks):

  1. Complete the truth table for the operation P → (Q ∧ R).
  2. Create a compound logical statement and construct its truth table.
  3. Prove whether P ↔ Q is true or false when both P and Q are false.
  4. Construct a truth table for the operation P ∨ (¬Q).
  5. Why is it important to use truth tables when analyzing logical statements?

 

PERIOD 5: Conditional Statements and Indirect Proofs

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Introduction

Introduces the concepts of conditional statements and their components: hypothesis and conclusion. Explains the converse, inverse, and contrapositive.

Students listen and ask questions about the components of conditional statements.

Step 2: Conditional Statements

Provides examples of conditional statements, such as "If it rains, then the ground will be wet."

Students observe and identify the hypothesis and conclusion in conditional statements.

Step 3: Converse, Inverse, and Contrapositive

Explains how to form the converse, inverse, and contrapositive of a given conditional statement.

Students work on forming the converse, inverse, and contrapositive of given statements.

Step 4: Indirect Proof

Introduces the concept of indirect proofs, explaining how to prove a statement by assuming its negation and reaching a contradiction.

Students observe the teacher’s demonstration of indirect proof and take notes.

NOTE ON BOARD:

  • Conditional Statement: "If P, then Q" (P → Q)
  • Converse: "If Q, then P" (Q → P)
  • Inverse: "If not P, then not Q" (¬P → ¬Q)
  • Contrapositive: "If not Q, then not P" (¬Q → ¬P)

EVALUATION (5 exercises):

  1. Identify the hypothesis and conclusion in the statement: "If it is cold, then I will wear a coat."
  2. Form the converse of the statement: "If it rains, then the ground will be wet."
  3. Form the inverse of the statement: "If I study, then I will pass the exam."
  4. Form the contrapositive of the statement: "If I eat too much, then I will feel sick."
  5. Provide an example of an indirect proof for a simple mathematical statement.

CLASSWORK (5 questions):

  1. Identify the hypothesis and conclusion in the statement: "If I go to the store, then I will buy milk."
  2. Write the converse of "If it is sunny, then I will go to the beach."
  3. Write the inverse of "If I wake up early, then I will have breakfast."
  4. Write the contrapositive of "If it is hot, then I will drink water."
  5. Demonstrate an indirect proof to show that "If a number is divisible by 4, then it is divisible by 2."

ASSIGNMENT (5 tasks):

  1. Prove the converse of the statement: "If a number is even, then it is divisible by 2."
  2. Write the inverse of the statement: "If I do my homework, then I will get good grades."
  3. Write the contrapositive of the statement: "If you are not careful, you will break the glass."
  4. Provide an indirect proof for the statement: "If a number is divisible by 3, then it is divisible by 6."
  5. Why is it important to understand the converse, inverse, and contrapositive in logical reasoning?