MIRROR LESSON PLAN

In the "Finding Patterns" section of the Beyond Numbers exhibit, visitors use floor tiles and 3-D structures in a hall of mirrors to explore such concepts as reflective symmetry, periodicity, and infinity. In the "Finding Patterns" section, visitors can experiment with slides, flips, turns, and glide reflections with strip patterns. There are also "explainer" activities around the museum, led by a docent, on reflections and quilts.

NCTM Standards and/or AAAS Benchmarks addressed:

Mathematics as Communication:
Students relate physical materials, pictures, and diagrams to mathematical ideas.

Number Sense and Numeration:
Students construct number meanings (infinity) through real-world experiences and the use of physical materials.

Patterns and Relationships:
Students recognize, describe, extend and create a wide variety of patterns.

The nature of Mathematics, Patterns and Relationships:
Students learn that mathematics is the study of many kinds of patterns, including numbers and shapes and operations on them. Sometimes patterns are studied because they help to explain how the world worked or how to solve practical problems, sometimes because they are interesting in themselves.

The Mathematical World, Shapes:
Students learn that many objects can be described in terms of simple plane figures and solids. Shapes can be compared in terms of concepts such as parallel and perpendicular, congruence and similarity, and symmetry. Symmetry can be found by reflection, turns or slides.

VOCABULARY

fundamental region -- the smallest part of a design that is repeated to form the entire design

line of symmetry -- line on which a mirror can be placed where it shows an image of the rest of the object

reflection -- a mirror image, also called a "flip"

translation -- a repetition of an image with the same up and down orientation, also known as a "slide," a composite of two reflections over parallel lines

rotation -- a repeated image that appears to be a turn of the original, also know as a "turn," a composite of two relfections over intersecting lines

glide reflection -- a combination of "slide" and "flip," a composite of reflection over vertical and horizontal lines.

INTRODUCTION

Physicists use concepts of symmetry to describe patterns that occur naturally in the structure of matter. Anthropologists can classify baskets, cloth and pottery by the type of symmetry transformations in their ornamentations. In the following activities students will create and analyze different kinds of symmetry transformations.

OBJECTIVES

Students will observe and describe transformations on a graphic design made by placement of mirrors.

MATERIALS

Crayons or magic markers, lined paper, assorted small objects about 1" long -- at least three of each kind and of three different kinds for each student, e.g. Cuisenaire rods, Multilink pieces, construction paper cut into shapes, transparent tape for each group of four students, 2" x 2" piece of reflecting material (mirrors or the mylar included with this manual) -- two for each student "reflectors" translucent devices such as MIRAs or REFLECTAs one for each student -- a substitute can be made with colored overhead projector acetate mounted on cardboard as shown.

PROCEDURE

Activity 1 - Lines of symmetry

Have students use a dark marker or crayon to print their names in upper case letters on a piece of paper. Ask "What will the name look like in a mirror?"

Have the students turn the paper over and trace the letters showing through.

Give every student a 2" x 2" piece of mirror material. Have them reflect their traced name in the mirror to see their name as it was originally. Tell students that they will be doing some activities that deal with patterns and reflections.

Ask students if some of the letters of their names looked the same when they were written in reverse. (W,T,Y,U,I,O,A,H,X,M) Point out that these letters have a vertical line of symmetry. Have them place their mirror vertically midway on the letters so that they can see where the line of symmetry is. Suggest that they may find other letters with another line of symmetry (horizontal). (E,I,O,D,H,K,X,B,C)

Give each student nine objects -- three each of at least three different kinds. Challenge them to manipulate the objects to form designs that have a line of symmetry. Then challenge them to form, designs that have no lines of symmetry.

Activity 2 - Symmetry types:

Have students make a design that has no line of symmetry. Guide students through the directions on Worksheets A and B (below)

For older children: Point out that there are four kinds of pattern repetition that makes a symmetrical pattern. The reflection or "flip", rotation or "turn," the translation or "slide," and the glide reflection, which is a combination of "flip" and "slide." The rotation is a composite of two reflections over intersecting lines; the translation is a composite of reflections done over parallel lines; and the glide reflection is a composite of reflections over horizontal and vertical lines. Thus, all of these symmetries or combinations of these symmetries can be done with reflections!

Activity 3 - Quilts:

With Worksheet C (below), students are given the fundamental regions of three traditional American quilt designs.

Let students color one design. Have them use two-, three-, and four mirror squares to see how multiple reflections affect the size of the pattern. With three mirrors, the pattern extends infinitely in two directions. Have students discuss their understanding of the meaning of infinite. With four mirrors, the pattern extends in all directions. Have students imagine stepping into a mirrored room such as the one they have made.

ASSESSMENT

Have students write narrative descriptions of a quilt design using shape names, placement terms such as "next to," "left," and descriptions of relationships such as "half," "smaller." Then have them rewrite the design description as it would be seen in one, two, three, and/or four mirrors.

WORKSHEET A

Symmetry types
TRANSLATION - MAKING A "SLIDE"

You will need: