Exploring Medians in Triangles

Objective:

Students will be able to identify, define, and apply the properties of medians in triangles, including understanding the concept of a centroid.

Assessment:

Students will complete a worksheet where they identify and draw medians in different triangles and explain the concept of a centroid.

Key Points:

• Definition of a Median in a Triangle: In a triangle, the line segment that joins a vertex and the midpoint of the opposite side is called a median.

• Centroid: If you draw all three medians they will intersect at one point called the centroid.

• Median Theorem: The medians of a triangle intersect at a point called the centroid that is two-thirds of the distance from the vertices to the midpoint of the opposite sides.

• Formula for Centroid: The coordinates of the centroid of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) are given by:
  ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

Opening:

• Engage students by asking: "Can you think of a scenario where finding the center point is crucial for stability or balance?"

Introduction to New Material:

• Explain the definition of a median in a triangle

• Discuss how medians intersect at the centroid

• Common Misconception: Students may think medians are angle bisectors

Guided Practice:

• Provide examples for students to identify and draw medians in triangles

• Scaffold questioning from easier examples to more complex cases

• Monitor student performance by circulating the classroom and providing feedback

Independent Practice:

• Assign a task where students need to calculate the centroid of a triangle given its vertices

• Encourage students to explain their reasoning and show all work

Closing:

• Summarize the concept of medians and centroids by having students draw and label a triangle with its medians

Extension Activity:

• For early finishers, provide a challenge where they explore the relationship between medians and other elements in triangles

Homework:

• Homework suggestion: Students research real-life applications of medians in construction or other fields and prepare a short presentation.

Standards Addressed:

CCSS.MATH.CONTENT.HSG.CO.B.7
CCSS.MATH.CONTENT.HSG.CO.B.7