A Tribute to M C Escher Tapestry Tessellations

A Tribute to M. C. Escher: Tapestry Tessellations Mac. Kinnon Middle School Seventh Grade Math Teachers

Introduction In this web quest you will be exploring regular polygons and their ability to tessellate. You will learn about the sum of angles as they relate to regular polygons. You will also explore the internal mathematical pattern found in the sum of angles formula and learn to identify regular polygons that tessellate together. In order to get a better understanding of the concept of tessellations, explore the following websites. Pay close attention to the information about the famous artist/mathematician, M. C. Escher. Carefully read the definition of tessellations on the second link and look at the examples. http: //library. thinkquest. org/11750/index. shtml (Escher) http: //mathforum. org/sum 95/suzanne/whattess. html (definition) www. coolmath 4 kids. com/tesspag 1. html

Links Activity • Summarize the definition of a tessellation • Identify the three regular polygons that tessellate alone • List three facts about M. C. Escher

Task The Metropolitan Museum of Art is planning a special tribute to honor the work of the famous artist/mathematician, M. C. Escher. The Met has announced a competition which will involve middle school students in designing a new tapestry pattern that will hang in the Escher exhibit in tribute to his work with tessellations. You and your partner have decided to enter this competition as the Met is offering a prize of $2, 000 for the winning design.

Process • Step 1: Choose a regular polygon tile provided by your teacher and carefully trace it on a sheet of blank paper. • Step 2: Extend the sides of the polygon so you can measure all the interior angles with a protractor. • Step 3: Calculate the sum of the interior angles. • Step 4: Complete the table indicating the following for each: – – – Name of polygon Number of sides Number of internal triangles Sum of angles Compute formula Identify whether the regular polygon will tessellate.

Process continued… • Step 5: Participate in a class discussion to identify the pattern formed by the sum of angles of regular polygons. Determine the number of triangles formed in each regular polygon. • Step 6: Identify the rules for polygons that tessellate. Discuss them with your partner or class. Notice that some regular polygons tessellate alone and some will tessellate with another regular polygon!

Process continued… • Step 7: Plan and design your tapestry of tessellating regular polygons with your partner. You can make your design using a range of materials: – draw the design, by hand or on the computer, and color it – create your design using pieces of colored paper, fabric, cardboard, etc. – create your own method of presenting your design

Process continued… • Step 8: Write a detailed description of your tapestry design. Include the following: – Type of polygon(s) used – Sum of the interior angles for each polygon used – Sum of angles at each vertex – Explain why your polygon(s) is /are able to tessellate.

Evaluation Finding a pattern _____x 2 = ____ Completing a table _____x 3 = ____ Tessellation tapestry _____x 1 = ____ Written description _____x 4 = ____ Novice = 7 Apprentice = 8 Practitioner = 9 Expert = 10 Students identify the sum of the angles of a triangle=180° Students identify sums of angles of regular polygons by measuring Students identify the sums of angles of seven regular polygons by discovering a pattern Students identify a formula for determining the sum of angles of regular polygons Table indicates name of polygon and number of sides Table indicates name of polygon, number of sides and number of internal triangles Apprentice plus the sum of the angles Practitioner plus the formula for the sum of angles correctly computed Some attempt to create a tessellated tapestry pattern is evident Student successfully used one regular polygon to create a tessellating tapestry pattern Successful use of two regular polygons to create a tessellating pattern Successful use of more than two regular polygons to create a tessellating pattern Includes two or three elements from step 8 of the process Includes sum of angles formula for each polygon, sum of angles at each vertex Apprentice plus tapestry dimensions and surface area Practitioner plus explanation of tessellating properties