Further Mathematics - Senior Secondary 1 - Logical reasoning

Logical reasoning

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TERM: 1ST TERM

WEEK 11
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Logical Reasoning
Focus: Rules of Logic Application to Argument; Implication and Deduction

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

  1. Understand the rules of logic and how they apply to mathematical and verbal arguments.
  2. Identify and explain the structure of logical implications (if…then… statements).
  3. Differentiate between antecedents and consequences.
  4. Apply logical deduction rules to solve problems and draw valid conclusions.

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Use of real-life examples
• Collaborative problem-solving
• Use of visual aids

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Charts showing conditional statements
• Flashcards with logical forms and examples
• Worksheets with logical reasoning exercises

 

PERIOD 1 & 2: Introduction to Logical Reasoning and Implication

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of logic and reasoning. Explains its importance in mathematics, computer science, and everyday thinking.

Students listen and relate logic to subjects like mathematics and computers.

Step 2 - Conditional Statements

Introduces the structure of a conditional statement (If P, then Q). Defines antecedent (P) and consequent (Q). Provides everyday examples.

Students give examples of “if…then…” statements from real life.

Step 3 - Truth Values

Explains how conditional statements can be true or false. Uses truth tables to show combinations of truth values for P and Q.

Students observe and attempt to fill in truth tables.

Step 4 - Implication

Demonstrates how implications work using examples: “If it rains, then the ground will be wet.” Explains how P → Q logic works.

Students participate in identifying P and Q from examples.

NOTE ON BOARD:

  • Conditional Statement: “If P, then Q”
  • P: Antecedent
  • Q: Consequent
  • Implication: P → Q
  • Truth Table for P → Q
    | P | Q | P → Q |
    |---|---|--------|
    | T | T | T |
    | T | F | F |
    | F | T | T |
    | F | F | T |

Students copy the table and notes from the board.

EVALUATION (5 exercises):

  1. Identify the antecedent and consequent in: “If I study, I will pass.”
  2. What does it mean for a conditional statement to be false?
  3. Complete the truth table for P → Q.
  4. Write your own conditional statement.
  5. Is the statement “If today is Sunday, then tomorrow is Monday” true or false?

CLASSWORK (5 questions):

  1. What are the two parts of a conditional statement?
  2. Identify the implication: “If water freezes, it becomes ice.”
  3. What is the truth value of “If 2 + 2 = 5, then 3 > 1”?
  4. Use a truth table to evaluate P → Q when P is true and Q is false.
  5. Write an implication involving school.

ASSIGNMENT (5 tasks):

  1. Write 3 examples of implication from daily life.
  2. Explain why “If P, then Q” can be true even when P is false.
  3. Construct a truth table for the implication “If it is hot, then I drink water.”
  4. Identify the antecedent and consequent in “If she is happy, then she sings.”
  5. Create a poster showing different conditional statements and their parts.

 

PERIOD 3 & 4: Deduction and Logical Problem Solving

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Deduction

Introduces the concept of deduction: drawing conclusions from known facts or premises. Gives simple examples.

Students listen and attempt deduction tasks.

Step 2 - Rules of Deduction

Explains key rules like modus ponens (If P → Q and P is true, then Q is true) and modus tollens (If P → Q and Q is false, then P is false).

Students copy the rules and apply them to examples.

Step 3 - Real-Life Application

Presents scenarios where students must use deduction to arrive at logical conclusions.

Students work in groups to discuss and solve deduction problems.

Step 4 - Practice

Gives structured exercises on implication and deduction with increasing difficulty.

Students solve exercises individually and in groups.

NOTE ON BOARD:

  • Deduction: Reasoning from known facts
  • Modus Ponens: If P → Q and P is true, then Q is true
  • Modus Tollens: If P → Q and Q is false, then P is false
  • Example: If it rains (P), the grass is wet (Q).
    • It rains → The grass is wet → Valid (Modus Ponens)
    • The grass is not wet → It didn’t rain → Valid (Modus Tollens)

Students copy rules and examples.

EVALUATION (5 exercises):

  1. Apply modus ponens to: “If I sleep early, I wake up early.” I slept early.
  2. Use modus tollens on: “If you eat sugar, you gain weight.” You didn’t gain weight.
  3. Identify if the deduction is valid: “If it’s cold, I wear a coat. I wore a coat.”
  4. Fill in the blank: If P → Q and Q is false, then _____ is false.
  5. Identify the deduction type used: “If she is tired, she yawns. She yawned.”

CLASSWORK (5 questions):

  1. Use deduction to solve: “If today is Monday, then we have assembly.” Today is Monday.
  2. Apply modus tollens: “If the bell rings, school is over. The bell didn’t ring.”
  3. Create your own example using modus ponens.
  4. Create a scenario using modus tollens.
  5. Identify whether this is modus ponens or modus tollens: “If the stove is on, it is hot. It is not hot.”

ASSIGNMENT (5 tasks):

  1. Construct two implications and apply deduction to them.
  2. Write three real-life examples of modus ponens.
  3. Write three real-life examples of modus tollens.
  4. Explain how logic is used in decision-making.
  5. Find a newspaper headline and write its logical form (If…, then…).