Mathematics - Senior Secondary 1 - Logical reasoning (I)

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of logical reasoning and the basic building block: the simple statement. Explains that a simple statement is a declarative sentence that is either true or false, but not both. Differentiates between open and closed statements with examples (e.g., "x is a prime number" vs. "2 is a prime number").

Students listen attentively and ask for clarification on the definitions of simple, open, and closed statements. They try to provide their own initial examples.

Step 2 - True or False Statements

Presents various closed simple statements and guides students to determine their truth value (true or false). Encourages justification for their answers. Examples: "Lagos is the capital of Nigeria." (False), "7 is greater than 3." (True).

Students analyze the given statements and state whether they are true or false, providing reasons for their decisions.

Step 3 - Negation of Simple Statements

Explains the concept of negation as the opposite of a statement. Demonstrates how to form the negation of a simple statement using words like "not" or the phrase "it is not true that". Provides examples and encourages students to negate given statements. Example: Statement: "The book is blue." Negation: "The book is not blue." or "It is not true that the book is blue."

Students learn the rules for forming negations and practice negating various simple statements provided by the teacher.

Step 4 - Guided Practice

Provides worksheets with simple statements and asks students to identify if they are open or closed and, if closed, determine their truth value. Also, asks them to write the negation of the given simple statements.

Students work individually or in pairs to complete the worksheet exercises. They discuss their answers and seek clarification from the teacher when needed.

NOTE ON BOARD

- Simple Statement: A declarative sentence that is either true or false. - Closed Statement: A statement that can be definitively determined as true or false. - Open Statement: A statement containing a variable that becomes true or false when the variable is replaced by a specific value. - Negation of a statement 'p' is denoted by '¬p' and has the opposite truth value of 'p'. Examples of simple statements and their negations.

Students copy the notes and examples from the board into their notebooks.

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Compound Statements

Introduces the concept of compound statements as statements formed by combining two or more simple statements using logical connectives. Focuses on the connectives "and" (conjunction) and "or" (disjunction). Explains the symbols used for these connectives (∧ for "and", ∨ for "or").

Students listen carefully and understand that compound statements are built from simple statements. They learn the symbols for conjunction and disjunction.

Step 2 - Conjunctions ("and")

Explains that a conjunction is true only when both of the simple statements it connects are true. Provides examples: "The sun is shining and it is warm." (True only if both are true). "2 is an even number and 3 is a prime number." (True). "The cat is black and the sky is green." (False). Guides students to form their own compound statements using "and" and determine their truth value based on the truth values of the simple statements.

Students observe the examples and understand the truth condition for a conjunction. They actively participate in forming their own conjunctive statements and determining their truth values.

Step 3 - Disjunctions ("or")

Explains that a disjunction is true when at least one of the simple statements it connects is true. Presents examples: "It is raining or it is sunny." (True if it's raining, sunny, or both). "4 is an odd number or 5 is a prime number." (True). "Lagos is in Ghana or Abuja is in Niger." (False). Clarifies the inclusive "or" (it can be both). Guides students to create disjunctive statements and determine their truth values.

Students understand the truth condition for a disjunction. They create their own disjunctive statements and analyze their truth values based on the truth values of the component simple statements.

Step 4 - Guided Practice

Provides worksheets with pairs of simple statements and asks students to form compound statements using "and" and "or". They then determine the truth value of each compound statement based on the truth values of the original simple statements.

Students work on the exercises, applying their understanding of conjunction and disjunction to form compound statements and evaluate their truth values. They discuss their reasoning with peers and the teacher.

NOTE ON BOARD

- Compound Statement: Formed by combining simple statements using logical connectives. - Conjunction (and, ∧): True only if both connected statements are true. - Disjunction (or, ∨): True if at least one of the connected statements is true. Examples of compound statements using "and" and "or" with their truth values.

Students copy the definitions, symbols, and examples of conjunction and disjunction from the board.

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Implication

Introduces the concept of implication (conditional statement) using the connective "if...then...". Explains the symbol for implication (⟹). Defines the antecedent (the "if" part) and the consequent (the "then" part). Provides examples: "If it rains, then the ground will be wet." Explains that an implication is only false when the antecedent is true and the consequent is false.

Students learn about implication and its symbol. They identify the antecedent and consequent in given examples. They understand the specific condition under which an implication is false.

Step 2 - Examples of Implication

Provides various examples of implications and guides students to identify the antecedent and consequent. Discusses the truth value of these implications in different scenarios. Examples: "If a number is divisible by 4, then it is divisible by 2." (True). "If a person lives in Nigeria, then they live in Europe." (False antecedent, so the implication is true). "If it is a fish, then it can fly." (True antecedent, false consequent, so the implication is false).

Students analyze the given implications, identify the antecedent and consequent, and determine their truth values based on the truth values of the component simple statements. They participate in discussions about why certain implications are true or false.

Step 3 - Introduction to Bi-implication

Introduces the concept of bi-implication (biconditional statement) using the connective "if and only if" (often abbreviated as "iff"). Explains the symbol for bi-implication (⟺). Explains that a bi-implication is true only when both connected statements have the same truth value (both true or both false). Provides examples: "A triangle is equilateral if and only if all its sides are equal."

Students learn about bi-implication and its symbol. They understand the condition for a bi-implication to be true (both parts having the same truth value).

Step 4 - Examples of Bi-implication

Provides examples of bi-implications and guides students to determine their truth values. Examples: "A number is even if and only if it is divisible by 2." (True). "The sun rises in the east if and only if the earth is flat." (False, because one part is true and the other is false).

Students analyze the given bi-implications and determine their truth values by considering the truth values of both component statements.

Step 5 - Guided Practice

Provides worksheets with pairs of simple statements and asks students to form compound statements using "if...then..." and "...if and only if...". They then determine the truth value of each compound statement.

Students work on the exercises, applying their understanding of implication and bi-implication to form compound statements and evaluate their truth values. They discuss their reasoning with peers and the teacher.

NOTE ON BOARD

- Implication (if...then..., ⟹): False only when the antecedent is true and the consequent is false. - Antecedent: The statement following "if". - Consequent: The statement following "then". - Bi-implication (if and only if, ⟺): True only when both connected statements have the same truth value. Examples of implications and bi-implications with their truth values.

Students copy the definitions, symbols, and examples of implication and bi-implication from the board.