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What to create	Lesson plan
Which subject	Mathematics
What topic	Biography, General Contributions and Contributions to parallel postulate of Legendre, Lambert and taurinus and Farkas bolya
What length (min)	30
What age group	College
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Lesson Plan: The Contributions of Legendre, Lambert, Taurinus, and Farkas Bolyai to the Parallel Postulate

Academic Subject: Mathematics

Duration: 30 Minutes

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Objectives

By the end of this lesson, students will be able to:

1. Understand the historical context and significance of the parallel postulate.
2. Discuss the contributions of Jean Étienne Montucla Legendre, Johann Heinrich Lambert, and others to the postulate.
3. Analyze the impact of their work on the development of non-Euclidean geometry.

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Lesson Outline

1. Introduction to the Parallel Postulate (5 Minutes)

• Brief explanation of the parallel postulate, one of Euclid's five postulates, which states that through a point not on a line, there is exactly one line parallel to the given line.
• Importance of this postulate in the framework of Euclidean geometry.

2. Biography of Key Contributors (10 Minutes)

Jean Étienne Montucla Legendre (1752-1833)

• Born in France, Legendre was a mathematician famous for his work in number theory and geometry.
• Significant contributions include the development of the Legendre polynomials and his book Éléments de Géométrie, where he reformulated the geometry without reliance on the parallel postulate.

Johann Heinrich Lambert (1728-1777)

• A German mathematician and astronomer whose experimentation with geometric concepts bordered on non-Euclidean perspectives.
• Known for his work Die Theorie der Parallellinien (The Theory of Parallel Lines), where he explored the complexities of the parallel postulate, suggesting alternatives.

Taurinus

• An English mathematician less known than Legendre and Lambert, but who contributed to discussions on geometry and the parallel postulate.
• His examinations brought into question the assumptions surrounding the foundations of geometric principles.

Farkas Bolyai (1802-1860)

• A Hungarian mathematician who, alongside his son János Bolyai, independently developed hyperbolic geometry.
• Farkas's work supported the theories that stemmed from questioning the parallel postulate and ultimately led to the acceptance of non-Euclidean geometries.

3. General Contributions to Mathematics (5 Minutes)

• Discuss how Legendre, Lambert, Taurinus, and Farkas Bolyai collectively advanced mathematical thought.
• The trend toward questioning the conventional forms of geometry paved the way for further exploration in mathematical theories.

4. Discussions and Group Activities (5 Minutes)

• Organize students into small groups to discuss how the contributions of each mathematician relate to modern geometric concepts.
• Encourage students to draw connections between historical perspectives and current mathematical practices.

5. Conclusion and Reflection (5 Minutes)

• Recap the importance of the parallel postulate and its implications on modern mathematics.
• Highlight how innovations by past mathematicians opened new avenues of exploration in geometry.

6. Homework Assignment (Assigned at the End of Class)

Tasks:

1. Write a one-page essay discussing the impact of Farkas Bolyai's contributions on the acceptance of non-Euclidean geometry.

2. Solve the following problems related to the parallel postulate:

   • Problem 1: Given a triangle where the sum of the angles is 200 degrees, classify the type of geometry it represents.

   • Problem 2: Prove or provide conditions under which two lines parallel to a third line in hyperbolic space diverge as they extend indefinitely.

Correct Answers:

1. Essay Response: Students should highlight that Bolyai's defense of hyperbolic geometry, which departed from the standard Euclidean framework, laid crucial groundwork for modern geometric systems and influenced future mathematicians like Einstein in the theory of relativity.

2. Problem 1: The triangle represents hyperbolic geometry, as in Euclidean geometry, the angles must sum to 180 degrees.

   Problem 2: In hyperbolic space, if two lines are drawn parallel to a third line from a point not on the line, the lines will diverge, consistent with the properties of hyperbolic geometry.

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End of Lesson Plan