Exploring Pythagorean Theorem and Pythagorean Triples

Objective:

Students will be able to apply the Pythagorean Theorem to find the missing side of a right triangle, identify Pythagorean Triples, and apply the Converse of the Pythagorean Theorem.

Assessment:

Design a worksheet where students are given multiple right triangles and are required to find the missing side using the Pythagorean Theorem. Additionally, include questions where students need to identify Pythagorean Triples and explain the Converse of the Pythagorean Theorem.

Key Points:

• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is: (a^2 + b^2 = c^2).

• Pythagorean Triples: These are sets of three positive integers that satisfy the Pythagorean Theorem. Some common Pythagorean Triples include 3-4-5, 5-12-13, and 8-15-17.

• Pythagorean Theorem Converse: This states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Opening:

• Engage students by asking, "How can we determine the length of the missing side in a right triangle?"

• Show a visual representation of the Pythagorean Theorem and ask students to explain what they observe.

Introduction to New Material:

• Present the Pythagorean Theorem formula and provide real-world examples for better understanding.

• Discuss the misconception of assuming any triangle is a right triangle just because it has a 90° angle.

Guided Practice:

• Begin with simple right triangle examples and gradually increase the complexity of the problems.

• Monitor students as they work through examples, providing guidance and clarification where needed.

Independent Practice:

• Assign a worksheet with a variety of right triangle problems involving the Pythagorean Theorem, Pythagorean Triples, and the Converse of the Pythagorean Theorem.

• Encourage students to show all steps of their work to demonstrate understanding.

Closing:

• Have students partner up and explain to each other how the Pythagorean Theorem can be used in finding missing side lengths in a right triangle.

Extension Activity:

• For early finishers, challenge them to research and present on other famous mathematical theorems related to geometry.

Homework:

• Create a homework assignment where students have to find and list Pythagorean Triples and provide an explanation for each.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.C.8.A: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

• CCSS.MATH.CONTENT.HSG.SRT.C.8.B: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.