Further Mathematics - Senior Secondary 1 - Logical reasoning

Logical reasoning

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TERM: 1ST TERM

WEEK 10
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Logical Reasoning
Focus:
i. The Truth Table
ii. Logical Connectives: "p or q" (p ∨ q), "p and q" (p ∧ q)
iii. Implication and Bi-conditional: (p ⇒ q), (p ⇔ q)
iv. Syntax, Simple True or False Statements

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

  1. Understand and construct truth tables for basic logical statements.
  2. Apply logical connectives to create compound statements.
  3. Interpret and use implication (p ⇒ q) and bi-conditional (p ⇔ q) in reasoning.
  4. Identify and analyze the truth value of simple statements.
  5. Solve logical reasoning problems using truth tables.

INSTRUCTIONAL TECHNIQUES:
• Guided demonstration
• Question and answer
• Group discussions
• Practice exercises
• Use of charts and logical puzzles

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Charts showing logical connectives and truth tables
• Flashcards with logical statements
• Worksheets for truth table construction

PERIOD 1 & 2: Introduction to Logical Reasoning and Truth Tables

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces logical reasoning with examples of simple true/false statements (e.g., “It is raining,” “2 + 2 = 5”). Defines statement and truth value.

Students listen and suggest examples of true or false statements.

Step 2 - Logical Connectives

Explains logical operations: OR (∨), AND (∧), implication (⇒), and biconditional (⇔). Uses real-life examples like “If I study, then I will pass.”

Students give examples of “or” and “and” statements.

Step 3 - Constructing Truth Tables

Introduces the format of truth tables. Guides students through creating truth tables for “p ∧ q” and “p ∨ q”.

Students create truth tables in their notebooks.

Step 4 - Syntax and Logical Form

Explains how to write logical expressions correctly. Emphasizes proper logical form and meaning.

Students practice forming simple logical expressions.

NOTE ON BOARD:
Logical Symbols:

  • p ∧ q → “p and q”
  • p ∨ q → “p or q”
  • p ⇒ q → “If p then q”
  • p ⇔ q → “p if and only if q”
    Truth Values: T = True, F = False

EVALUATION (5 exercises):

  1. Write two examples of simple true/false statements.
  2. What is the meaning of the symbol “∨”?
  3. Construct the truth table for p ∧
  4. Construct the truth table for p ∨
  5. What is the truth value of “2 is an odd number”?

CLASSWORK (5 questions):

  1. Fill in the truth table for p and q, and p ∨
  2. Fill in the truth table for p ∧
  3. Evaluate the statement: “If today is Monday, then tomorrow is Tuesday.”
  4. Is the statement “All birds can fly” true or false?
  5. Construct a truth table for: p ∨ (¬q)

ASSIGNMENT (5 tasks):

  1. Write a logical expression for: “I go out if it is not raining.”
  2. Construct a truth table for p ⇒
  3. Construct a truth table for p ⇔
  4. Identify the truth value of: “0 is greater than 1.”
  5. Research one real-life use of logical reasoning.

 

PERIOD 3 & 4: Implications, Biconditionals and Practical Application

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Implication (p ⇒ q)

Explains implication with examples: “If I sleep early, I will wake up early.” Shows truth table for implication.

Students take notes and ask questions.

Step 2 - Biconditional (p ⇔ q)

Introduces bi-conditional, explains that it is true only when both statements have the same truth value.

Students copy truth table and give examples.

Step 3 - Application Exercises

Solves example problems involving compound statements and truth tables.

Students work in pairs to solve practice problems.

Step 4 - Syntax Rules

Reviews proper syntax in logical expressions and identifies common student errors.

Students correct poorly structured logical statements.

NOTE ON BOARD:
Truth Table for p ⇒ q

p

q

p ⇒ q

T

T

T

T

F

F

F

T

T

F

F

T

Truth Table for p ⇔ q

p

q

p ⇔ q

T

T

T

T

F

F

F

T

F

F

F

T

EVALUATION (5 exercises):

  1. Construct a truth table for p ⇒
  2. Construct a truth table for p ⇔
  3. What does the statement p ⇒ q mean in English?
  4. Determine the truth value of: If 3 is even, then 5 is prime.
  5. Which connective means “p is true if and only if q is true”?

CLASSWORK (5 questions):

  1. Construct a truth table for: (p ∧ q) ⇒ p
  2. Construct a truth table for: ¬p ∨ q
  3. Translate into logic: “You pass if you study.”
  4. Identify the connective: ⇔
  5. Write the logical form of: “If I eat, I will not be hungry.”

ASSIGNMENT (5 tasks):

  1. Construct a truth table for: (p ∨ q) ⇔ (¬p ∧ q)
  2. Write 2 real-life statements as implications.
  3. Create a compound statement using p and q.
  4. Research how truth tables are used in computer science.
  5. Explain the difference between p ⇒ q and p ⇔