Lesson 6: Inscribed Quadrilaterals

Introduction

Student Audience:

This lesson is designed for implementation with high school students studying Euclidean Geometry. Completing the activities in this lesson is anticipated to require 1.5 one-hour class periods.

Objectives:

· Students create numerous quadrilaterals including the following and determine if they can be inscribed in a circle: square, rectangle, parallelogram, rhombus, trapezoid, kite, irregular quadrilateral

· Students will note whether particular quadrilaterals can be inscribed in a circle and record properties of these quadrilaterals, such as angle measures and side lengths.

· Students will organize data using either a chart or electronic spreadsheet

· Students will analyze data, determining the properties that determine whether a quadrilateral can be inscribed in a circle.

· Students will generalize their findings and develop an if-and-only-if statement about the properties of inscribed quadrilaterals.

· Students will use formal proof to verify their conjecture.

Mathematical Concepts:

· Types of quadrilaterals and associated properties (square, rectangle, parallelogram, rhombus, kite, trapezoid, irregular quadrilateral)

· Inscribed quadrilateral

· Data collection and organization

· Supplementary angles

· Measure of an inscribed angle

· Property of arc addition

· Sum of interior angles of a quadrilateral

· Degrees in a circle

Lesson Synopsis:

Using Geometer’s Sketchpad software, students will construct familiar quadrilaterals and determine whether each can be inscribed in a circle. Students will collect data about the properties of each of these quadrilaterals. Students will then expand their search of quadrilaterals that can be inscribed in a circle by manipulating the Geometer’s Sketchpad file provided. Students can examine and collect data about the properties associated with irregular quadrilaterals that can be inscribed in a circle. Students will then analyze their data to form conjectures about what quadrilateral properties determine whether a quadrilateral can be inscribed in a circle. After articulating their findings, students will use formal proof to confirm their conjectures.

Lesson

Materials:

· Geometer’s Sketchpad software

· DynamicSketch.gsp Geometer’s Sketchpad file

· Spreadsheet software (optional)

· Student Data Organization Table (optional)

Implementation:

The instructor will begin the lesson by reviewing the concept of inscribed polygons. The instructor may choose to focus on previous work with inscribed triangles or may extend the discussion to other contexts. Ultimately, the instructor should focus students on the fact that, for a polygon to be inscribed in a circle all vertices of the polygon must lie on the same circle. The instructor will then pose the question of what kinds of quadrilaterals can be inscribed in a circle. Students may be encouraged to make conjectures about familiar quadrilaterals such as squares and rectangles. The instructor will then encourage students to utilize the Geometer’s Sketchpad software to construct familiar quadrilaterals and attempt to inscribe them in a circle. During this component of the lesson, students should collect data about each quadrilateral they investigate including attributes ( side length, angle measure) and whether it could be inscribed.

After students investigate these quadrilaterals, the instructor will lead the class in a brief discussion of their findings. The instructor may use the facilitator questions provided to highlight the similarities and differences among these quadrilaterals. Following this brief discussion, the instructor will pose the question of whether any irregular quadrilaterals can be inscribed in a circle. The instructor will direct students (either individually, or in small groups) to manipulate the DynamicSketch.gsp file and try to find any irregular quadrilaterals that can be inscribed in a circle. For any inscribable irregular quadrilaterals, students should record prominent attributed (side lengths, angle measures, presence of any parallel sides, presence of any perpendicular sides). A sample collection of data can be found here.

As students accumulate data, they should analyze their findings looking for patterns and ultimately properties that determine whether a quadrilateral can be inscribed in a circle. During this portion of the investigation, the instructor will circulate the room, addressing concerns and incorporating facilitator questions as appropriate. Students may become overwhelmed by the amount of data that they are collecting and consequently, the instructor should be cognizant of offering supportive, but not directing, advice.

When students have developed a conjecture the instructor should guide them toward developing a proof of their conjecture. By utilizing the notion that the measure of and inscribed angle is half the measure of its intercepted arc and the fact that the arcs intercepted by opposite angles form a complete circle, students should be able to prove that the opposite angles of any inscribed quadrilateral must be supplementary.

Facilitator Questions:

· What does it mean for a polygon to be inscribed in a circle?

· Compare the situation in which a polygon is inscribed in a circle and that same polygon has a circumscribed circle.

· What are the properties of common quadrilaterals? Compare and contrast these quadrilaterals.

· Which quadrilaterals do you than can be inscribed in a circle? Why?

· What quadrilateral properties do you think would be important to record data about for our investigation?

· What properties do the common quadrilaterals that can be inscribed in a circle have?

· Do you think that it is possible for an irregular quadrilateral to be inscribed in a circle? Why?

· Were you able to find any irregular quadrilaterals that could be inscribed in a circle? How many?

· Did you see any common properties among inscribable quadrilaterals?

· How can examples help you form a conjecture?

· How can examples discredit a conjecture?

· When would a proof be useful for supporting a conjecture?

· When examining an example of an inscribed quadrilateral, what other elements of circle geometry can you find?

· How might these other elements be useful for your developing a proof of your conjecture?

NAME _______________________________ DATE __________________

Student Data Organization Table

	Inscribe?
Square
Rectangle
Parallelogram
Rhombus
Kite