Students will be able to construct and complete two-column proofs by providing statements and reasons in a logical sequence.
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.
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.
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.
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.
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.
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.
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.
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.
For homework, students should complete a set of additional two-column proofs provided in the textbook or worksheets.
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.