Understanding Segments from Chords

Objective:

Students will be able to apply the Intersecting Chords Theorem to find the lengths of segments created by chords intersecting inside a circle.

Assessment:

Students will demonstrate their understanding by solving problems involving segments from chords within circles.

Key Points:

• Chords: Segments that connect two points on a circle.

• Segments within a Circle: Portions of a circle divided by chords.

• Intersecting Chords Theorem: If two chords intersect within a circle, the product of the segments of one chord is equal to the product of the segments of the other chord. This can be represented by the formula: AB * BC = CD * DE, where AB and BC are segments of one chord and CD and DE are segments of the other chord.

Opening:

• Engage students by displaying a diagram of a circle with intersecting chords and asking: "How can we determine the length of segments created by these chords?"

Introduction to New Material:

• Define chords and segments within a circle

• Explain the Intersecting Chords Theorem and its application

• Anticipated misconception: Students may confuse the lengths of segments and chords within a circle.

Guided Practice:

• Provide examples of intersecting chords for students to work through

• Scaffold questioning from basic calculations to more complex problem-solving

• Monitor student progress by circulating the room and providing guidance as needed

Independent Practice:

• Assign a series of problems involving segments from chords

• Ensure students show their work clearly to demonstrate understanding

• Provide additional support for students who may need it

Closing:

• Have students share their responses to the opening question to summarize the key learning points of the lesson.

Extension Activity:

• Early finishers can explore real-world applications of segment lengths within circles, such as in architecture or design.

Homework:

• Homework activity suggestion: Create a set of circle diagrams with intersecting chords and ask students to calculate segment lengths based on the given information.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG-C.A.4: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.

• CCSS.MATH.CONTENT.HSG-C.A.4b: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.