Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: First Term
Week: 11
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Logical Reasoning
Previous Lesson: Logical Reasoning: Meaning, Intelligent Systems and Modelling the World.
Topic: LOGICAL REASONING
Subject Matter: Introduction to propositional logical resolution, introduction to predicate logical resolution, introduction to theorem proving

Specific Objectives

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

Cognitive Domain:

Define propositional logic and predicate logic.
Identify logical connectives and quantifiers.
List the steps involved in proving a theorem.

Affective Domain:

Appreciate the importance of logical reasoning in problem-solving.
Show interest in using logical methods to analyze statements.

Psychomotor Domain:

Apply propositional and predicate logical resolution to simple statements.
Demonstrate simple steps in proving a theorem.

Discuss different logical statements and arguments with peers.
Collaborate to solve basic logical puzzles.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics for Senior Secondary Schools by P.N. Okeke and others.
https://www.mathsisfun.com/logic/propositional-logic.html
https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_propositional_logic.htm

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing points to note in proving of theorem
Whiteboard or chalkboard
Markers or chalk
Handouts with simple logical problems

Rationale for the Lesson

This lesson enables pupils to understand the fundamental principles of logical reasoning, which is important for critical thinking and problem-solving in mathematics and everyday life. It helps pupils to develop analytical skills needed to evaluate statements and construct valid arguments.

Prerequisite/Previous Knowledge

Pupils are expected to have a basic understanding of statements, true and false values, and simple algebraic expressions.

Lesson Content/Board Summary

LOGICAL REASONING

Introduction to Propositional Logic

Propositional logic deals with propositions (statements) that can be either true or false. It uses logical connectives to combine simple propositions into more complex ones.

A proposition is a declarative sentence that is either true or false, but not both.

The following are common logical connectives:

Conjunction (AND): Symbolised by $land$. $P land Q$ is true if and only if both P and Q are true.
Disjunction (OR): Symbolised by $lor$. $P lor Q$ is true if P is true, or Q is true, or both are true.
Negation (NOT): Symbolised by $neg$. $neg P$ is true if P is false, and false if P is true.
Implication (IF…THEN): Symbolised by $to$. $P to Q$ is false only when P is true and Q is false.
Biconditional (IF AND ONLY IF): Symbolised by $leftrightarrow$. $P leftrightarrow Q$ is true if P and Q have the same truth value.

Introduction to Predicate Logic

Predicate logic extends propositional logic by allowing statements to contain variables and quantifiers. It provides a way to express properties of objects and relationships between them.

A predicate is a property that a variable in a universe of discourse can have. For example, in “x is even”, “is even” is the predicate.

The following are common quantifiers:

Universal Quantifier (FOR ALL): Symbolised by $forall$. $(forall x) P(x)$ means “for all x, P(x) is true”.
Existential Quantifier (THERE EXISTS): Symbolised by $exists$. $(exists x) P(x)$ means “there exists an x such that P(x) is true”.

Introduction to Theorem Proving

Theorem proving is the process of demonstrating that a mathematical statement (theorem) is true based on a set of axioms or previously established theorems. A common method is direct proof.

Steps in Direct Proof:

Assume the hypothesis (the “if” part) is true.
Use definitions, axioms, and previously proven theorems to logically deduce the conclusion (the “then” part).
State the conclusion.

Worked Example 1: Propositional Logic

Construct the truth table for $(P land Q) to neg Q$.

Step 1: List all possible truth values for P and Q.

Step 2: Calculate the truth values for $P land Q$.

Step 3: Calculate the truth values for $neg Q$.

Step 4: Calculate the truth values for $(P land Q) to neg Q$.

P	Q	$P land Q$	$neg Q$	$(P land Q) to neg Q$
T	T	T	F	F
T	F	F	T	T
F	T	F	F	T
F	F	F	T	T

Worked Example 2: Theorem Proving (Direct Proof)

Prove that if $n$ is an even integer, then $n^2$ is an even integer.

Step 1: Assume the hypothesis. Let $n$ be an even integer.

Step 2: By the definition of an even integer, $n$ can be written as $n = 2k$ for some integer $k$.

Step 3: Square $n$: $n^2 = (2k)^2 = 4k^2$.

Step 4: Rewrite $4k^2$ as $2(2k^2)$. Let $m = 2k^2$. Since $k$ is an integer, $k^2$ is an integer, and $2k^2$ is also an integer. So, $m$ is an integer.

Step 5: Thus, $n^2 = 2m$, where $m$ is an integer. By the definition of an even integer, $n^2$ is an even integer.

Step 6: Conclude. Therefore, if $n$ is an even integer, then $n^2$ is an even integer.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher introduces the lesson by asking pupils to identify if simple sentences are true or false, such as “The sun rises in the west” or “2+2=5”.
Pupils’ Activity: Pupils respond to the teacher’s questions, stating whether the sentences are true or false.
Learning Point: Pupils are engaged and introduced to the concept of truth values in statements.

Step 2: Introduction to Propositional Logic

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines propositional logic and explains what a proposition is. The teacher then introduces logical connectives (AND, OR, NOT, IMPLIES, BICONDITIONAL) with their symbols and basic truth conditions, using examples.
Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and note down definitions and symbols.
Learning Point: Pupils understand propositions and the basic logical connectives.

Step 3: Introduction to Predicate Logic

Time: 10 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains predicate logic as an extension of propositional logic. The teacher defines predicates and introduces quantifiers (Universal and Existential) with their symbols and meanings, using simple examples like “all students are smart” or “some birds can fly.”
Pupils’ Activity: Pupils follow the explanations, identify predicates and quantifiers in given sentences, and ask questions.
Learning Point: Pupils grasp the concepts of predicates and quantifiers in logical statements.

Step 4: Introduction to Theorem Proving

Time: 10 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher explains the concept of a theorem and the importance of proving them. The teacher outlines the general steps for a direct proof and demonstrates a simple example from the board summary, such as proving that the sum of two even numbers is even.
Pupils’ Activity: Pupils pay attention to the steps and the worked example, noting down the procedure for proving theorems.
Learning Point: Pupils understand the basic process of theorem proving and can follow simple proof steps.

Step 5: Practice Exercise

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher gives pupils a short exercise to determine the truth value of a compound proposition or to rephrase a quantified statement.
Pupils’ Activity: Pupils attempt the exercise individually or in pairs.
Learning Point: Pupils apply their understanding of propositional and predicate logic.

Step 6: Review and Clarification

Time: 5 minutes
Teaching Skill: Question and Answer
Teacher’s Activity: The teacher reviews the answers to the practice exercise and addresses any difficulties or misconceptions pupils may have.
Pupils’ Activity: Pupils share their answers and ask further questions for clarification.
Learning Point: Pupils consolidate their understanding and clarify any doubts.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define a proposition.
List any three logical connectives and their symbols.
Explain the difference between a universal quantifier and an existential quantifier.
State the first two steps in proving a theorem by direct proof.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the importance of logical reasoning in mathematics and daily life. The teacher assigns homework, asking pupils to find examples of propositions and predicates from their textbooks.
Pupils’ Activity: Pupils listen to the summary, ask final questions, and copy down the homework.
Learning Point: Pupils reinforce their learning and prepare for further practice.

Lesson Keywords

Proposition – A declarative sentence that is either true or false.
Predicate – A property that a variable can have.
Connectives – Symbols like AND, OR, NOT used to combine propositions.
Quantifiers – Symbols like FOR ALL ($forall$) and THERE EXISTS ($exists$) used in predicate logic.
Theorem – A mathematical statement that can be proven to be true.
Proof – A logical argument that establishes the truth of a theorem.

Differentiation

For pupils who are struggling, the teacher will provide simpler examples and more direct guidance during problem-solving. Advanced pupils will be challenged with more complex logical statements and proofs, encouraging them to explore different proof techniques.

Note for teachers using this lesson plan

Teachers should ensure that pupils understand the basic concepts of truth values before introducing connectives and quantifiers. Encourage active participation through questions and examples from real-life scenarios. Emphasize the systematic approach to theorem proving, as it is a foundational skill in higher mathematics.