Exploring Indirect Proof in Algebra and Geometry

Objective:

Students will be able to use indirect proof to prove statements by assuming the opposite and reaching a contradiction.

Assessment:

Students will be given a set of statements to prove using indirect proof. They must correctly assume the opposite of the conclusion, find a contradiction, and state the original statement is true.

Key Points:

• Indirect proof: An indirect proof takes the conclusion from a hypothesis and assumes it is false until a contradiction is reached, thus proving the original hypothesis is true.

• The steps to follow when proving indirectly are:

  • Assume the opposite of the conclusion (second half) of the statement.

  • Proceed as if this assumption is true to find the contradiction.

  • Once there is a contradiction, the original statement is true.

• Indirect proof involves using variables instead of specific examples for generalization.

• Contradictions play a crucial role in proving statements indirectly.

Opening:

• Introduce the concept of indirect proof using a real-world scenario.

• Engage students by asking: "How can we prove something to be true by assuming it's false?"

Introduction to New Material:

• Explain the process of indirect proof with step-by-step guidance.

• Anticipate the misconception that assuming the opposite conclusion is enough to disprove a statement.

Guided Practice:

• Provide examples for students to work on in pairs.

• Scaffold questioning from basic assumptions to complex contradictions.

• Monitor students as they work through the process, offering guidance when needed.

Independent Practice:

• Assign students a set of statements to prove using indirect proof.

• Expect students to show their reasoning and contradictions clearly in their work.

Closing:

• Have students share their findings from the independent practice.

• Summarize the key steps of indirect proof and its significance in proving statements.

Extension Activity:

• Challenge early-finishers to create their own statements to prove using indirect proof, exchanging with a partner for solving.

Homework:

• Ask students to research and write a brief explanation of a real-world situation where indirect proof could be applied.

Standards Addressed:

• CSS.MATH.CONTENT.HSG.CO.D.14: Prove theorems about parallelograms.

• CSS.MATH.CONTENT.HSG.CO.D.14: Prove theorems about triangles.