Mathematics - Senior Secondary 1 - Logical reasoning (II)

Logical reasoning (II)

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TERM: 2ND TERM

WEEK: 5

Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes × 5 periods
Subject: Mathematics
Topic: Logical Reasoning (II)
Focus: Logical operations and symbols – Truth value table – Compound statement, Negation, Conditional statement, Bi-conditional statement.

 

SPECIFIC OBJECTIVES:

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

  1. Identify five basic logical operations and their corresponding symbols.
  2. Construct and interpret truth tables for each logical operation.
  3. Understand and form compound logical statements.
  4. Apply negation, conditional, and bi-conditional operations.
  5. Analyze and evaluate compound statements using truth values.

 

INSTRUCTIONAL TECHNIQUES:

  • Guided discussion
  • Socratic questioning
  • Truth table demonstrations
  • Peer collaboration
  • Concept mapping

 

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Printed charts of truth tables
  • Flashcards with logical symbols and statements
  • Worksheets for truth table exercises

 

PERIOD 1 & 2: Introduction to Logical Operations and Symbols

PRESENTATION:

Step

Teacher’s Activity

Students’ Activity

Step 1 - Introduction

Introduces the idea of logical reasoning in mathematics and real-life decision-making.

Students listen and provide real-life examples of logical decisions.

Step 2 - Logical Operations

Lists and explains the five logical operations: Negation (¬), Conjunction (∧), Disjunction (∨), Conditional (→), Bi-conditional (↔).

Students note down the operations and their meanings.

Step 3 - Symbols and Statements

Demonstrates how symbols are used to represent logical statements. For example, “p: It is raining”, “q: The ground is wet”.

Students observe and translate given statements into symbolic form.

Step 4 - Truth Values

Explains the concept of truth values (True = T, False = F). Demonstrates simple truth value combinations.

Students practice evaluating the truth value of simple statements.

NOTE ON BOARD:

Logical Operations:

- Negation (¬p)

- Conjunction (p ∧ q)

- Disjunction (p ∨ q)

- Conditional (p → q)

- Bi-conditional (p ↔ q)

Truth Values: T (True), F (False)

 

EVALUATION (5 exercises):

  1. What symbol is used for negation?
  2. Write the symbolic form of: “If I study, I will pass.”
  3. What are the possible truth values of a statement?
  4. What is the opposite of “p is true”?
  5. Which logical operation uses the symbol ∧?

 

CLASSWORK (5 questions):

  1. Translate “It is sunny or it is cold” into symbols.
  2. Identify the logical operation: “p ↔ q”.
  3. What is the negation of: “I will travel”?
  4. Write a compound statement using ∨.
  5. How many truth values does a single logical statement have?

ASSIGNMENT (5 tasks):

  1. Define each logical operation and give one real-life example.
  2. Represent “If I eat, then I am full” symbolically.
  3. What does the symbol ↔ represent?
  4. Write two compound statements using ∧ and ∨.
  5. Explain why truth values are important in logic.

 

PERIOD 3 & 4: Constructing Truth Tables for Logical Operations

PRESENTATION:

Step

Teacher’s Activity

Students’ Activity

Step 1 - Truth Table Concept

Introduces the truth table and explains its structure using two statements: p and q.

Students listen attentively.

Step 2 - Table for Conjunction & Disjunction

Draws and explains the truth table for p ∧ q and p ∨ q using T and F combinations.

Students copy and practice constructing the tables.

Step 3 - Table for Conditional & Bi-conditional

Constructs truth tables for p → q and p ↔ q with full explanation.

Students analyze the patterns and interpret results.

Step 4 - Guided Practice

Gives students exercises to complete partially filled truth tables.

Students complete and present their work for correction.

NOTE ON BOARD:

Truth Table for p ∧ q:

p | q | p ∧ q

T | T | T

T | F | F

F | T | F

F | F | F

 

Truth Table for p → q:

p | q | p → q

T | T | T

T | F | F

F | T | T

F | F | T

 

EVALUATION (5 exercises):

  1. What is the truth value of T ∧ F?
  2. What is the truth value of T ∨ F?
  3. What is the truth value of F → T?
  4. What is the result of ¬F?
  5. How many rows are in the truth table for two statements?

 

CLASSWORK (5 questions):

  1. Fill in the missing values:
    p = T, q = F; what is p → q?
  2. What is the result of ¬T?
  3. Complete the table for p ∨ q when p = F and q = F.
  4. Create a truth table for p ↔ q.
  5. List all possible combinations of truth values for two statements.

 

ASSIGNMENT (5 tasks):

  1. Complete a full truth table for p ∧ q and ¬p.
  2. What is the output of ¬p ∨ q when p = T, q = F?
  3. Construct the truth table for ¬p → q.
  4. Describe a situation where p ∧ q must both be true for success.
  5. What does it mean when p ↔ q is false?

 

PERIOD 5: Analysis of Compound Statements Using Truth Tables

PRESENTATION:

Step

Teacher’s Activity

Students’ Activity

Step 1 - Introduction

Reviews truth tables and introduces compound statements combining more than one operation (e.g., ¬p ∨ q).

Students follow and ask clarifying questions.

Step 2 - Example Analysis

Demonstrates with an example: (¬p ∨ q), p = T, q = F → ¬p = F, F ∨ F = F.

Students take notes and follow the steps.

Step 3 - Evaluation Practice

Provides several compound statements and guides the class through evaluating each using truth tables.

Students practice individually and in pairs.

Step 4 - Class Interaction

Students share their truth table analysis and reasoning.

Students actively participate in presenting and correcting solutions.

 

NOTE ON BOARD:

Compound Statements Example:

p = T, q = F

¬p = F

¬p ∨ q = F ∨ F = F

 

Evaluate compound statements using:

  1. Negation (¬)
  2. Truth table logic
  3. Proper substitution

 

EVALUATION (5 exercises):

  1. Evaluate: p = T, q = T → What is ¬p ∧ q?
  2. Determine: p = F, q = T → What is p ∨ ¬q?
  3. Compute: p = T, q = F → What is ¬p → q?
  4. State the meaning of “compound statement”.
  5. How many operations are in the statement ¬p ∨ q?

 

CLASSWORK (5 questions):

  1. Construct the truth table for ¬p ∧
  2. Evaluate: p = F, q = F → What is p ↔ q?
  3. Evaluate the truth value of ¬(p ∧ q) when both are T.
  4. Interpret: “If I am hungry, then I eat” in symbols.
  5. Give one example of a compound logical statement from real life.

 

ASSIGNMENT (5 tasks):

  1. Create your own compound logical statement and evaluate it using a truth table.
  2. Explain the role of negation in changing truth values.
  3. Why is truth table analysis important in logic?
  4. Evaluate: (¬p → q) when p = F, q = T.

Construct the truth table for p ∧ (¬q).