Understanding Two-Column Proofs

Objective:

Students will be able to construct and complete two-column proofs by providing statements and reasons in a logical sequence.

Assessment:

Students will be assessed through a worksheet where they will be required to complete several two-column proofs independently, providing both statements and reasons for each step.

Key Points:

• Understanding the structure of a two-column proof

• Differentiating between statements and reasons

• Applying deductive reasoning to complete proofs

• When writing your own two-column proof, keep these things in mind:

  • Number each step.

  • Start with the given information.

  • Statements with the same reason can be combined into one step. It is up to you.

  • Draw a picture and mark it with the given information.

  • You must have a reason for EVERY statement.

  • The order of the statements in the proof is not always fixed, but make sure the order makes logical sense.

  • Reasons will be definitions, postulates, properties and previously proven theorems. “Given” is only used as a reason if the information in the statement column was given in the problem.

  • Use symbols and abbreviations for words within proofs.

Opening:

• Engage students by presenting a real-world scenario that requires logical reasoning to solve, such as proving why a certain car will reach its destination in a given time.

• Ask students how they think mathematicians prove their theories or relationships in geometry.

Introduction to New Material:

• Explain the concept of two-column proofs and show examples to illustrate the format of statements and reasons.

• Emphasize the importance of clarity and logical sequence in constructing proofs.

• Address the common misconception that every step in a proof must relate directly to the step before it.

Guided Practice:

• Provide guided examples of constructing two-column proofs, starting with simple scenarios and gradually increasing complexity.

• Scaffold questioning by first asking students to identify statements and reasons separately, then guiding them to combine them into a logical progression.

• Monitor student performance by circulating the classroom, providing feedback, and intervening where necessary.

Independent Practice:

• Assign students a set of two-column proofs to complete independently, varying in difficulty to ensure mastery.

• Encourage students to articulate their thinking process by explaining each step clearly.

• Specify that both statements and reasons must be included for full credit.

Closing:

• Conduct a quick recap session where students share one key concept they have learned about two-column proofs.

• Summarise the importance of logical reasoning and clear communication in mathematical proofs.

Extension Activity:

Challenge early finishers to create their own two-column proofs based on given theorems or geometric relationships. Encourage them to swap with a partner and attempt to solve each other's proofs.

Homework:

For homework, students should complete a set of additional two-column proofs provided in the textbook or worksheets.

Standards Addressed:

• CCSS Standard: G-CO.A.12 - Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

• CCSS Standard: G-CO.A.13 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.