Understanding Conjectures and Counterexamples

Objective:

Students will be able to differentiate between conjectures and counterexamples, and identify counterexamples to disprove given conjectures.

Assessment:

Students will be given a set of statements and will need to provide counterexamples for each to disprove the conjectures.

Key Points:

• Understanding what a conjecture is

• Defining a counterexample

• Identifying and creating counterexamples

• Differentiating between conjectures and counterexamples

Opening:

• Introduce the concept of conjectures and counterexamples through real-life examples

• Pose a question to students: "Can you think of a time when you made an assumption that turned out to be incorrect?"

Introduction to New Material:

• Define conjectures and counterexamples clearly

• Provide examples of conjectures and ask students to explain why they think it's a conjecture

• Anticipate common misconception: Students may initially struggle to differentiate between the two concepts

Guided Practice:

• Present students with a series of conjectures and guide them in creating counterexamples to disprove each one

• Begin with simple examples and gradually increase the complexity

• Monitor student performance by circulating the classroom, providing support, and offering feedback

Independent Practice:

• Instruct students to come up with their own conjectures and test them with counterexamples

• Provide a worksheet with statements for students to identify and create counterexamples

• Encourage students to work individually and apply their understanding

Closing:

• Conduct a brief class discussion to summarise the key differences between conjectures and counterexamples

• Ask students to share one conjecture they thought was true but was disproved by a counterexample

Extension Activity:

• Challenge early finishers to create a conjecture without a counterexample and share it with the class for discussion

Homework:

• Homework activity suggestion: Ask students to find examples in daily life where making a conjecture led to incorrect assumptions and share them in the next class.

Standards Addressed:

• CCSS Standard: G-GPE.B.5 - Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems

• CCSS Standard: G-GPE.B.5 - Construct a function to model a linear relationship between two quantities