## Reflect On This

- Lesson

This lesson, adapted from an activity in *Navigating through Geometry in Grades 9‑12*, requires students to investigate reflections using hinged mirrors. With a kaleidoscope, students will examine the interior angles of regular polygons.

A kaleidoscope creates fascinating designs that change as the kaleidoscope is rotated. Two mirrors hinged together may be used to model the effects of a kaleidoscope.

Have each student create a model of a kaleidoscope, following these steps:

- Place the reflective sides of two mirrors face-to-face. Tape one set of edges together to make a hinge.
- Cut small pieces of confetti from colored paper.
- Overlap a sheet of white paper with a sheet of colored paper. Half of each sheet should remain visible.
- Position the hinged mirrors across the two sheets of paper, as shown below. Place the hinged mirror so that AB ≈ AC.

To see a video of this, right click here and choose "Save Link As" to download the video, then launch it from your desktop.

*Still can't see the video? Click here to install Apple's free QuickTime player. *

Allow students a minute or two for free investigation. Let them see what happens when they change the angle of the mirrors or the arrangement of the confetti and paper.

- Open and close the mirrors, keeping AB ≈ AC. Observe the patterns formed by the confetti and colored paper.
- Make a record of the patterns formed by the colored paper.
- Rearrange the confetti, then open and close the mirrors. Observe the new patterns formed.

Distribute the Reflect On This Activity Sheet, and have them complete the questions on the sheet. Circulate and answer questions while students are working. In addition, use this time to assess the level of student understanding, and provide assistance as necessary.

Reflect On This Activity Sheet

When students have completed the activity sheet, conduct a class discussion by asking the following questions.

- Describe the patterns created by the colored paper in your kaleidoscope.
- What happened as you opened and closed the hinged mirrors?
- What is the relationship between the size of the hinge angle and the number of reflections seen?

### Reference

Day, Roger, Paul Kelley, Libby Krussel, Johnny W. Lott, and Jame Hirstein. Navigating through Geometry in Grades 9‑12. Reston, VA: NCTM. 2001.

- Mirrors (about 10 cm × 12 cm)
- Tape
- Sheets of white and colored paper
- Protractors
- Straightedges
- Reflect On This Activity Sheet
- Reflections Activity Sheet

**Assessment Option**

Students should be able to transfer their discoveries from this lesson to application problems. To assess their level of understanding, distribute the Reflections Activity Sheet, and allow students to complete it as an in-class assignment or as homework. To follow up, discuss the answers to the activity sheet as a class.

### Learning Objectives

Students will:

- Use multiple reflections to create regular polygons.
- Determine the relationship between the number of sides in a regular polygon and the measure of its central angles.
- Examine the significance of congruent angles and their applications.

### NCTM Standards and Expectations

- Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.

- Use various representations to help understand the effects of simple transformations and their compositions.

- Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.