Time Allotment: Five 50-minute class periods.

Overview: The study of transformations - translations, reflections, and rotations - provides the mathematics student with insight into many standard mathematics problems in algebra, geometry and pre-calculus. Through the activities in this lesson, students will use hands-on activities to explore transformations. They will view a video segment that will demonstrate how M. C. Escher employed geometry and transformations to create so many of his famous drawings, and then they will investigate how a kaleidoscope uses transformations and mirrors to create beautiful patterns. As a culminating activity, students will use the results of their investigation to build their own kaleidoscope.

Learning Objectives: Students will be able to:

I. Number Sense and Operations: Compute fluently and make reasonable estimates.

II. Patterns, Relations, and Algebra

1. Understand patterns, relations, and functions.

2. Use mathematical models to represent and understand quantitative relationships.

III. Geometry

1. Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships.

2. Apply transformations and use symmetry to analyze mathematical situations.

IV. Measurement

1. Understand measurable attributes of objects and the units, systems, and processes of measurement.

2. Apply appropriate techniques, tools, and formulas to determine measurements.

10.N.3, 4, 7

10.G.1, 2, 3, 5, 6, 9

10.M.1, 2, 4

10.P.11

12.G.5

Media: The Fantastic World of M.C. Escher (Atlas Video Inc.)

Kaleidoscopes!

HyperStudio & Tesselations in Color

Escher in the Classroom - Classroom Activities

For each pair of students:

*Cut out pattern block for each student - this can be any asymmetrical shape , ex. a fish with one fin.

*Cue the video tape.

*Prior to the instruction of this material, bookmark the Web sites used in the lesson. Load in any plug-ins necessary to run the Web sites.

*Set up the overhead projector.

*Make copies of Worksheets # 1 - # 7 for each student.

*The set of instructions for construction of a kaleidoscope are dependent upon the materials that your students plan to use. Visit the Web site above to prepare a set of instructions appropriate for your class.

*Gather kaleidoscope materials not collected by the students.

Congruent Figures -- figures that have the same size and shape

Image -- a figure that results from a transformation

Line of Reflection -- a line in a plane that lies equidistant from any two corresponding points in a figure that

has reflective symmetry

Line of Symmetry -- a line of reflection is called a line of symmetry whenever the reflection image of every

point in the figure about this line is also a point of the figure.

Reflection -- a transformation that mirrors a figure across a given line; a flip

Regular Polygon -- a polygon with all its sides congruent and all its angles congruent

Rotation -- a transformation that turns a figure about a given point; a turn

Tessellation (plane) -- a covering of an infinite plane, without any gaps or overlaps, by a pattern of one or

more congruent shapes

Transformation -- (in this lesson) a movement of a figure to a new location, leaving the figure unchanged in

size and shape

Translation -- a transformation that moves all the points of a figure the same distance and direction; a slide

without a rotation

Step 1: Introduce the students to a display of several items (scraps of wallpaper or wallpaper borders with geometric patterns, copies of M. C. Escher prints, etc.) that were created using transformations. Moving from item to item, ask the students to focus on one particular pattern in each item and try to identify the movements that were used to create the overall pattern. Encourage students to associate the movements they see in the patterns with the words slide, flip and turn. Explain to the students that these movements are called transformations and that the use of transformations is one way in which many artists have used mathematics to create their art. Continue to explain that in mathematics the slide, flip and turn are referred to as translation, reflection and rotation.

Step 2: If possible, assign students to work in pairs. Provide each student with at least one sheet of unlined paper, writing instrument, pattern block, and straight edge. Provide each group of students with several crayons or colored pencils.

Using an overhead projector, place a pattern block on a transparency and trace around it. Have the students place their pattern block in a similar position on the unlined paper and trace around their pattern block as well. Then slide the pattern block to demonstrate the movement of a translation and trace around the pattern block in its new position. Have the students repeat these steps using their pattern block. Remove the pattern blocks from the overhead and unlined paper so that the students can view the effect of the translation. Have the students color the images the same color and then label this pair of drawings as a translation using the color code they have chosen.

Repeat these steps to create a reflected image using the piece of spaghetti as the line of reflection on the overhead while students use a straight edge to construct a reflection line on their paper. Have the students select a different color for this transformation.

Repeat again to create a rotated image, again selecting a different color for this transformation. Note here that reflections can also be considered a rotation if the original image is symmetric.

Each student should now have a labeled example of the three types of transformations - a translation, reflection and rotation. (Note: Students should select a different color code for each transformation.) This sample sheet will serve as a resource when completing the practice worksheets if they need reminders as to the definition of any of the three transformations.

Step 3: Ask students to complete worksheet #1 to re-enforce their understanding of the definitions of translation, reflection, and rotation. Have pairs of students compare their answers. Check student work.

Step 4: Explain to students that there is an easier way to draw reflections with the use of a geometric tool called a mira. Explain that this tool is both reflective like a mirror and transparent like glass so that they will be able to look through the mira to see a reflection and at the same time be able to trace the image that they see on the other side of it. Before distributing the miras, remind students to handle them carefully so that the surfaces do not get scratched and lose their reflective and transparent quality.

Provide each student with a mira and worksheets #2, #3, and #4. Using worksheet #2, have students use the mira to draw reflection images of figures by following these steps:

(a) place the mira on the line of reflection

(b) carefully trace the image that they see behind the mira.

When students are comfortable using the mira, have them use it to solve the word puzzles on Worksheet #3.

Worksheet #4 can be used as class work or homework to check the students' understanding of lines of symmetry. When the worksheet is completed, have students use miras to check their results.

Step 1: Show the class a few examples of the work of M. C. Escher. Ask if anyone is familiar with his work. Students can describe the type of work they have seen and in what context - perhaps on posters, t-shirts, calendars, etc. For those unfamiliar with him, explain that he was an artist who used transformations to create very interesting and sometimes mysterious drawings and that the class will be using the internet to become more familiar with him and his work.

Distribute the Focus for Media Interaction Worksheet (Worksheet #5).

Focus for Media Interaction: To provide students with a specific task to complete while viewing, ask them to complete the "Focus for Viewing Worksheet" (Worksheet #5) Part I as they are viewing the site. This activity will provide them with biographical information on Escher and allow them to view his many styles of work. Remind students to look carefully for the use of transformations in the creation of his work.

Ask students to log on to The World of Escher - Artwork Gallery

Focus for Media Interaction: To provide students with a specific task to complete while viewing, ask them to complete the "Focus for Media Interaction" Worksheet (Worksheet #5) Part II as they are viewing the site. This activity will provide them with the opportunity to view more of Escher's artwork and to learn how each of the works shown was created.

When the students have completed the viewing activities, orally review the students' responses to the questions on the "Focus for Media Interaction" Worksheet (Worksheet # 5).

Step 3: Explain to students that they will now watch a video that will analyze Escher's use of transformations in his work. To provide them with a specific task to complete while viewing provide the students with a copy of the "Focus for Viewing Worksheet" (Worksheet #6). Tell the students that they will be responsible for completing the worksheet while watching the video. Read through the questions with the students before the video begins. Remind students that they may ask to "pause, " "stop," or "rewind," the video at any point in time if they need time to jot down an answer or to view a segment for a second time.

Insert The Fantastic World of M.C Escher into your VCR. Cue the tape.

Begin the video as the image of Escher fills the window of the porthole (there will be no audio at this point, only a musical interlude). Pause the video as the narrator says "others that are very simple," and ask students to describe the patterns that they see in the mosaics. Encourage them to look for any transformations that might have been used to construct the patterns. Ask them to jot these down on their "Focus for Viewing" Worksheet.

Resume play once the students have answered the question. Pause the tape as the speaker says "...the atoms are arranged more or less in the same way. They have the same properties of symmetry and repetition as it is in these pictures of the Alhambra." Ask the students "What properties of the Alhambra mosaics are also found in the study of crystals and atoms?" Have the students jot their answer down on the Focus for Viewing Worksheet.

Resume play once the students have answered the question. Pause when the speaker says "... he wanted to make tessellations with figures that one could recognize as living figures." Ask the students what it was that Escher did not like about the Alhambra mosaics and how he changed that in his own drawings of mosaics?

Have the students write down their answer on their "Focus for Viewing" Worksheet.

Resume play once the students have answered the question. Pause as the speaker says "... pronounced interest in the tessellation of the cobblestone and the roof and the houses and everything." Ask the students to explain what early evidence we have that Escher was interested in structure and tessellations. How did this evidence appear in his work? Ask the students to write down their answer on their "Focus for Viewing Worksheet"

Resume play once the students have answered the question. Pause as the speaker says "... the black birds are the mirror image of the white birds and Escher was amused by this statement..." Ask the students why they think Escher was amused by the statement in a scientific magazine that suggested that the black and white birds were mirror images of each other. If necessary, rewind and resume play so that the students can view the picture again. Have them write their conclusion on their "Focus for Viewing" Worksheet.

Resume play once the students have completed their answer. Near the end of the video segment, there will be a series of tessellations drawn by Escher. Pause briefly on each of the following images and ask the students to identify the type of transformation - translation, reflection, or rotation that they observe in each one. Ask them to make a note on the transformation of each tessellation on their "Focus for Viewing Worksheet". Pause briefly on each of the following images: white/green/red lizards; white/blue geese; white geese/blue fish; multi-colored fish; knights on horses.

Stop the video as the image of the knights on horses fades from view.

Post-Viewing Activity: After viewing the video, review the questions on the "Focus for Viewing Worksheet" and their responses. In conclusion, have the students offer their own interpretation of the term tessellation and guide them to a simple definition of the form " a covering of an infinite plane, without any gaps or overlaps, by a pattern of one or more congruent shapes."

Step 4: Explain to students that they will now create tessellating patterns using three mirrors.

Provide each pair of students with a series of three mirrors (a pair of hinged mirrors and a third single one is best, if available) and a copy of the "Kaleidoscope/Tessellation Investigation Worksheet"(Worksheet # 7). Have the students work in pairs to perform the tessellation investigation and answer the questions on the worksheet.

Discuss the results of their investigation. Particularly, have the students note the difference between the design created when two mirrors are used and the design created when three mirrors are used - two mirrors generate a circular design while three generate an infinite design (tessellation). If necessary, have the students repeat the investigation of one of the patterns using two mirrors and then three mirrors to see this difference. Explain that these concepts provide the basis for the construction of a kaleidoscope.

Culminating Activity: Explain to the students that they will now design and build their own

Step 1. Use an overhead transparency to show the students a diagram of a typical kaleidoscope - a sample of a diagram is given on the web site noted above. Use this diagram or prepare one of your own. Note that the type of kaleidoscope that is shown and the type that they will construct require that the whole scope be rotated for the colored pieces to create different designs.

Explain the parts of the kaleidoscope. Have students brainstorm ideas for the items on the materials list that they will need to build their kaleidoscope. Encourage them to use inexpensive, found or recycled materials if at all possible.

Suggestions for materials:

*Glass end pieces - recycled clear plastic from rigid bakery and deli containers.

*Tube - potato chip canisters, Crystal Light containers, cardboard tubes used to hold Gestetner

stencils, paper towel tubes

* Mirrors - silver contact paper from a recycling center, silver wrapping paper, or recycle the inside packaging from a box of snack crackers such as Wheatables - it is silvery and very reflective. All of these can be mounted on strips of poster board or 'tablet backs' to create inexpensive mirrors.

*"Stuff" to view - beads, bits of glass, foil, ribbon, etc.

*Covering for tube - wallpaper scraps, paint, construction paper, wrapping paper

Step 2. Have students make a list of materials that they need to gather for each component of the kaleidoscope. If necessary, help students locate materials. Designate a time period during which the materials will be collected, (ex. one week) and select the day for kaleidoscope construction. Encourage students to help each other in gathering the materials.

Step 3. Have students make a drawing of the construction plan for their kaleidoscope, indicating the dimensions and materials to be used.

Step 4. On construction day, provide students with a list of steps to follow when constructing their kaleidoscope. Go over the steps in order with the students before they begin actual construction. Provide them with scissors, rulers, newspapers to protect work surfaces, access to a glue gun and/or spray adhesive, and any items that you arranged to gather for them. Have students work in pairs if possible. (Measuring, gluing, etc. will be more easily done in pairs.)

Construct the kaleidoscopes. Be prepared to assist with any problems that might occur during the actual construction. (The Web site mentioned above offers many tips for solving such situations.)

Step 5. Have show-and-tell time with the finished kaleidoscopes.

Creative Arts/Mathematics: Have students create circular kaleidoscopic designs on paper. KALEIDOSCOPIC DESIGNS and How to Create Them by Leslie G, Finkel, Dover Publications, Inc., NY, 1980 is an excellent resource. Even students who are not very artistic will be able to create intricate kaleidoscopic designs using the step by step instructions and patterns provided in this resource.

Creative Arts/History: Have students research the history of the kaleidoscope and the many styles that have been constructed.

Mathematics/Science: Have students build a life-size model of a three-mirror kaleidoscope out of recycled materials. Such a kaleidoscope would be large enough for a student to stand inside the mirror system to see first hand the effect of a three-mirror kaleidoscope.

Mathematics/Science: Have students investigate the symmetric properties of snowflakes and other patterns in nature and then prepare a display of the results of their investigation.

1. Invite a local artist/craftsman who designs and builds kaleidoscopes to speak to the class. Have the students prepare a list of questions that they would like to ask regarding the construction of his/her kaleidoscopes. If at all possible, take the students on a field trip to visit the craftsman's workshop and view the kaleidoscopes under construction.

2. Invite a local museum curator or kaleidoscope collector to speak to the class about the history of kaleidoscopes. If there is a collection of kaleidoscopes in a nearby museum, arrange a field trip for the students to view the collection.

3. Invite a local graphic artist to speak to the class, or if possible, arrange a field trip for the class to a graphic arts studio. Be sure to ask how the artist uses the concepts of transformations and symmetry when creating his/her designs.

"Transformations Worksheet" #1

"Reflections Using a Mira Worksheet" #2

"Solving Word Puzzles Worksheet" #3

"Lines of Symmetry Worksheet" # 4

Worksheets #1 - #4 are taken from the Addison-Wesley Informal Geometry Laboratory Manual.

"Focus for Media Interaction Worksheet" #5

"Focus for Viewing Worksheet" #6

"Focus for Media Interaction Worksheet"#7