Tessellations and Symmetry 8 th Grade Math

1. Tessellations and Symmetry8th Grade Math Duoc Nguyen Karen Miller Cara Williams Educ 3308-02 May 5, 2004 Assignment #11-Planning for Reflective Teaching

2. National Standards: National Council of Teachers of Mathematics Geometry Standards Grades 6-8: standards 1 through 4. Process Standards Grades 6-8: Problem Solving, Reasoning and Proof, Communication, Connections, and Representation

3. State Quality Core Curriculum Topic: Edge, Face, Vertex, n-gon Standard: “Classifies plane and solid geometric figures based on their properties / characteristics (number or length of sides, angle measures, edges, faces, or vertices). This includes the family of quadrilaterals and triangles (acute, obtuse, right, equilateral, isosceles, scalene);… and n-gons (pentagon, hexagon, octagon).”

4. State QCC’s • Topic: Similarity, Congruence Standard: “Uses properties to determine similarity and congruency of geometric figures.” • Topic: Geometric Properties Standard: Uses geometric figures, properties, and relations to solve problems • Topic:Reflection, Rotation, Translation Standard: Analyzes effects of basic transformations on geometric shapes.

5. Class Profile Racial Diversity East Cobb Middle School

6. Class Profile Criterion-referenced Competency Tests 8th grade math Socio-economic Standards

7. Class profile • The students have a diverse set of learning styles and preferences, multiple intelligence and ability levels. • The lesson is designed to appeal different aspects of learning intelligence: Visual/Special by doc cam. Verbal/Linguistic by teacher. Music/Rhythmic by patterns. Bodily/Kinesthetic by hands on activities. Interpersonal/Intrapersonal by group work. Logic/Mathematical are right at home.

8. Lesson Objectives • The students will individually discover, after the teacher’s demonstration of tessellations, which regular polygons tessellate by tracing different shapes and applying the given definitions. • The class will, after a class discussion, recognize examples and non-examples of line and rotational symmetry by interpreting examples presented by the instructor. • After learning about the artist M.C. Escher, the students will be able to recall and analyze Escher’s drawings and identify the different symmetries used in his artwork during the in class discussion and on the homework. • The students will, after interpreting Escher’s artwork, apply their knowledge of tessellations and symmetry by constructing artwork of their own. • The students will, after the discussions, activities and homework, have a better understanding of why symmetry and tessellations are important by getting into small groups and researching the use of tessellations in nature and man-made objects and presenting their findings to the class.

9. Reflectional Symmetry also called line symmetry, mirror symmetry or bilateral symmetry: If a figure can be reflected over a line in such a way that the resulting image coincides with the original. Example: the letter T Rotational Symmetry: if a figure can be rotated about a point in such a way that it’s rotated image coincides with the original figure after turning less than 360 degrees then the figure has rotational symmetry. Examples: letter Z Lesson Introduction-Symmetry Review

10. Translational Symmetry: if a pattern can be translated a given distance in a given direction in such a way that the image coincides with the original, then the patter has translational symmetry. Examples: show pictures Glide-reflection Symmetry: If a design can undergo a glide reflection isometry in such a way that the image coincides with the original then the design has glide reflection symmetry. Examples: the leaf Lesson Introduction-Symmetry Review

11. M.C. ESCHER Show some Escher art work and ask the children to brainstorm the connection between these pictures and symmetry. After having a question and answer period, have the students research tessellations on the internet.

12. Lesson Procedures • Identifiable teaching methods: Lesson will use the constructivist teaching method. The discovery approach of the constructivist teaching method will be used to identify which regular polygons tessellate and which do not as well as which semi- regular polygon combinations tessellate. Variety of technology including geometry sketchpad, the doc cam, Tessellation Station (an internet resource) websites, and power point. The teacher will also be using some direct instruction methods including lecturing and explaining, seatwork and homework.

13. Lesson Procedures • Implementation Procedures: After reviewing the types of symmetry, use this power point to define and explain tessellations beginning with the regular polygon. Use geometry sketchpad to demonstrate the symmetries. Hand out the student packet of regular polygons. Each child will receive different shapes of regular polygons. Instruct the students to trace repetitive shapes and discover which shapes tessellate and which do not. Allow ten minutes for this activity. Question the students to see which shapes tile and which did not. Return to tessellation station on the internet and build the triangle and the hexagon tessellation. Demonstrate how the pentagon does not tessellate.

14. Lesson Procedure Return to power point and begin the explanation of semi-regular tessellation. Access Tessellation Station and build a semi-regular tessellation. Assign for homework the discovery of the eight semi-regular tessellations. Introduce Escher’s art work and tessellations. Show the students Escher’s art and explain which symmetry was used. Use the document cam to show how the translations and rotations are made. Hands on tessellations using a note card. Begin this assignment and make sure the children understand the process. Their completed project will be due in one week

15. Lesson Procedure • The final connection is where Tessellations are found in nature. Use the doc cam to explain that the honeycomb uses a hexagon which creates the maximum volume with the least amount of effort. • In addition mention that snakes scales and kernels of corn are very near tessellations. Discuss man made tessellations. Materials needed Discovery Activity: Cardboard cutouts of a regular triangle, square, pentagon, hexagon, octagon, and decagon, one of each, per student or group. Escher Activity: One per student, Note card, tape, scissors, patty paper

16. Teacher Technology: Microsoft Word and websites were used to prepare this lesson. Geometers Sketchpad, Power Point and Tessellation Station were used to present this lesson. Student Technology: The student’s used the internet to research, mathematical software including: -Maple 8 - Geometer’s Sketchpad and Pascgalois -TI-83 calculator. Implementation of Technology

17. Resources • TEACHER: 1. a textbook used in high school 2. Tessellation Station: a website used to create regular and semi-regular tessellations 3. book on M.C. Escher artwork 4. online search on tessellations and symmetry in nature and man- made objects 5. tessellation discoveries using Geometer’s Sketchpad software • STUDENT: 1. Tessellation Station online 2. Geometer’s Sketchpad 3. books on M.C. Escher 4. research of tessellations in nature by using encyclopedias, science journals, books, etc.

18. Identifiable Motivational Strategies • The students at the 8th grade level are beginning to focus more on intrinsic motivation. They have the belief in their ability to succeed. The teacher should provide the students with an atmosphere where their answers will not be graded. With the opportunity to discover tessellations and symmetry during the hands-on tracing activity, the students are motivated by their individual success and the low threat of judgment from the teacher. • Some students may not have access to computers and/or the Internet in their homes. So, when assigning the group research project, the students are able to use the school’s computers and Internet access motivating the students to research their topics.

19. Identifiable Motivational Strategies 1. The artwork assignment (creating tessellations like M.C. Escher) allows students to explore their artistic abilities in an environment (a math classroom) that they wouldn’t usually do such projects. The motivation is the creative outlet. 2. Artwork project will be displayed in the lunchroom where classmates and other teachers can view their work. The social recognition the students will receive by this display will motivate the completion of the artwork assignment. 3. By completing the homework and participation in class, students will be extrinsically motivated by a reward. Fun Friday every month. Students can only participate if all assignments were competed and handed in on time.

20. Informal and formative Assessment: Students will engage in discussion regarding their research on Tessellations as well as their discoveries of regular and irregular tessellations. I will observe the students discovery lesson to ensure that they understand. Formal and Summative Assessment: The students will have three specific assignments for this lesson. They will finish the tessellation discovery lesson for homework. They will also have the Escher note card project that will be due one week after it is assigned. There is also a homework assignment that includes incorporating what they have learned and demonstrating a knowledge using three types of technology. Assessment

21. 1. Learning Styles: Visual Learners Auditory Learners Kinesthetic Learners Tactile Learners Active Reflective Global Understanding Analytical Understanding 2. Learner Characteristics: Verbal/Linguistic Music/ Rhythmic Bodily/ Kinesthetic Visual/ Spatial Interpersonal Intrapersonal Logic/ Mathematical Average ability Post Lesson Reflection 3.Cultural and Social Characteristics: 36% Black 39% white 9.4% Hispanic 7.1 % Asian 0.2 American Indian 7.8 Multiracial High and low SES 33% free lunch

22. 4. Learning Styles Visual Learners learn by observation. (Escher art & Power point). Auditory Learners benefit from the teacher taught portion of the lesson as well as the group work. The Kinesthetic Learners benefit from demonstration and from teaching others what they know. Tactile Learners learn by touching and manipulating. The hands on activities are very good for these students. The Active Learners enjoyed the reflection exercise where they got to actively experiment. The Reflective Learners benefited by getting a long term project they can think about. The Global Learners prefer the connection between the math, the real world applications and the art work. (See big picture) Analytical Understanding Learners see three trees first and benefit from the different steps. 5. Learner Characteristics The classroom profile this lesson has been prepared for is an eighth grade class room that is an inclusion classroom. This is a general education population where the students have a diverse set of learning styles and preferences, multiple intelligences and ability levels. The lesson is designed to appeal visual/special intelligence by projecting images using the doc cam. Verbal/Linguistic Intelligence will benefit from the teacher taught lesson and discussion. Music/Rhythmic Intelligence learners think in sounds rhythms and patterns and this lesson is all about patterns. There are two hands on activities for the Bodily and Kinesthetic Learners. The Interpersonal and Intrapersonal learners will be able to work with others and of course the Logic/Mathematical Learners are right at home in a math class. Post Lesson Reflection

23. 6. Cultural and Social Characteristics This lesson discusses art, math tessellations found in nature, and man made tessellations. It appeals to a wide range of economic profiles and ethnicities. Tessellations can be found in every day life for all income levels and exists in all cultures, from tiling to brick work 7. Motivational Strategies The students at the 8th grade level are beginning to focus more on intrinsic motivation. They have the belief in their ability to succeed. By creating a comfortable learning environment, students feel less stress in their performances. Also, their artwork is displayed for all to see giving the students the self esteem boost they might be searching for. While individual success is a key motivator, middle schoolers still like to have the extrinsic motivations. These motivations are met by scheduling a Fun Friday for the students who turned in their assignments on time. Post Lesson Reflection

24. Post Lesson Reflection 8. Bloom’s Taxonomy • Knowledge: the students should remember shapes and polygons and have a general understanding of what it means to be symmetric. They should also know how to research items in a library and online. • Comprehension: during the in class lecture, the students should begin to understand the axioms of symmetry and what a tessellation is. • Application: By tracing different shapes, the students apply the axioms to discover which n-gons tessellate. The students will also look outside the math classroom and identify different types of symmetry and tessellations. • Analysis: By analyzing different artwork by M.C. Escher, students should recognize the types of symmetry being used. • Synthesis: The students will then have the opportunity to use what they have learned and created symmetric artwork of their own. Evaluation: The students will be evaluated formally through tests, quizzes, and homework. Informally, the students will be evaluated by their individual and group participation, and also by the completion of their artwork

25. Post Lesson Reflection • 9. Appropriate Objectives The objectives planned for the in-class assignments are group assignments. Each activity will have ample in class time for the middle school learner to get a firm understanding on the topic and expectations so that when the student gets home, he/she will be able to complete the assignment with minimal questions. This technique entails directive and peer guidance. These objectives embrace learning as discovery and build upon the idea that students at this age enjoy the feeling of success and can very well achieve this success through proper explanations and exciting projects. Some of the appropriate challenges that the students will face will be applying the definitions of symmetry learned in class to their art and group projects.

26. 10. Introduction and Student Participation The introduction does a brief review of symmetry and an introduction to Tessellations. The discussion of Escher’s artwork is designed to capture the student’s attention and get the discussion going. There are many optical illusions in Escher’s artwork that will make which will also engage the students in discussion 11. Teaching Methods The constructivist method of teaching is effective in holding the interest if a wide range of learners. It gives children a hand- on discovery experience that they will retain much longer than more traditional approached • 12. Learning Activities • The learning activities planned for the in-class assignments are mostly group assignments. Each activity will have ample in class time for the middle school learner to get a firm understanding on the topic and expectations so that when the student gets home, he/she will be able to complete the assignment with minimal questions. This technique entails directive and peer guidance. These activities embrace learning as discovery and build upon the idea that students at this age enjoy the feeling of success and can very well achieve this success through proper explanations and exciting projects. Some of the appropriate challenges that the students will face will be applying the definitions of symmetry learned in class to their art and group projects Post Lesson Reflection

27. 13. Technology Resources Maple 8 is an effective way to demonstrate mathematical models in three dimensions where on paper previously we have only had two and to do complicated calculations. Pascalgalois is an excellent way to show symmetry using manipulations of Pascal’s triangle. Geometry Sketchpad allows the student to manipulate the symmetries in both rectangular and polar coordinates. The doc cam in the math and science building is awesome. I hope that I will have one in my classroom 14. Active Learning The resources for this lesson range from books to the Internet, and they each support active learning and participation from the class. Students will be able to actively search the library to check out resources for their projects. They will also use an online program to create virtual tessellations allowing them to gain a better understanding of the symmetry axioms. Geometer’s Sketchpad software will give the students a hands-on approach to different types of symmetry. Post Lesson Reflection

28. Post Lesson Reflection 15. Assessment • We used a wide variety of assessments that will allow different types of learners to excel. We also tried to make our assessments low stress by having many different methods in addition to testing. Our hope is that each student will find some of the activities interesting and motivating, encouraging children to succeed. The End