Unravelling Deductive Reasoning

Objective:

Students will be able to apply deductive reasoning rules such as the Law of Detachment, Law of Contrapositive, and Law of Syllogism to draw conclusions based on given facts.

Assessment:

Students will be given a set of statements and will be tasked with identifying the correct application of deductive reasoning laws to draw valid conclusions.

Key Points:

• Law of Detachment: If the hypothesis p → q is true and p is true, then q is true. In other words, if "if p then q" is true and p is true, then q must also be true.

  • Formula: ( (p \rightarrow q) \land p \therefore q )

• Law of Contrapositive: If the conditional statement p → q is true, then the contrapositive ~q → ~p is also true. This means if the conclusion q is false, then the hypothesis p is false.

  • Formula: ( p \rightarrow q \therefore \neg q \rightarrow \neg p )

• Law of Syllogism: If p → q and q → r are true statements, then p → r is also a true statement. This law helps in connecting the implications of two conditional statements.

  • Formula: ( (p \rightarrow q) \land (q \rightarrow r) \therefore p \rightarrow r )

Opening:

• Engage students in a scenario where they have to solve a mystery using deductive reasoning.

• Pose a question like: "If all cats are mammals, and Mittens is a cat, what can we deduce?"

Introduction to New Material:

• Explain deductive reasoning and its importance in logical thinking.

• Present examples illustrating the Law of Detachment, Law of Contrapositive, and Law of Syllogism.

• Anticipate the misconception that deductive reasoning always leads to a correct conclusion.

Guided Practice:

• Provide practice problems for students to apply the deductive reasoning laws.

• Scaffold questioning from simple to complex to deepen understanding.

• Monitor student progress by observing their problem-solving approaches.

Independent Practice:

• Assign a set of logical reasoning problems for students to solve independently.

• Ensure the assignment aligns with the mastery of deductive reasoning rules.

• Clearly outline the expected format for presenting their conclusions.

Closing:

• Have students share key conclusions they drew using deductive reasoning.

• Summarise the importance of deductive reasoning in real-life situations.

Extension Activity:

Students who finish early can create their own deductive reasoning problems for a peer to solve.

Homework:

Ask students to observe situations in their daily lives where deductive reasoning could be applied, and jot down the conclusions they draw.

Standards Addressed:

• CCSS Standard: G-GPE.B.6 - Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

• CCSS Standard: G-GPE.B.6 - Use coordinates to prove simple geometric theorems algebraically.