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Puzzling Tessellations Content Standards and Objectives Content Standard 2: Algebra (MA.S.2) Students will:

- demonstrate understanding of patterns, relations, and functions Objectives:

- A.1.2.5 analyze a given set of data for the existence of a pattern numerically, algebraically, and graphically

Content Standard 3: Geometry (MA.S.3) Students will:

- analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships

- apply transformations and use symmetry to analyze mathematical situations. Objectives:

- G.3.14 develop properties of tessellating figures and use those properties to tessellate the plane. Best Practices for Instruction and Instructional Strategies Activity: Students will:

- Research tessellations on the web. (See handout. Be sure all students have an acceptable use policy on file.)

- Create patterns that will tessellate a plane. - Draw and color tessellations to be used as a jigsaw puzzle. All puzzle pieces will have the same size

and shape.

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- Make ‘connections’ between the puzzle pieces so that there is only one way in which the puzzle can be solved.

o These ‘connections’ can be: o a math rule, definition, or question, o a visual or algebraic pattern, o a math problem, o related to school spirit.

- Cut out their puzzle pieces. - Share their puzzles with two or three other classmates. (as time permits) - Critique each others’ puzzles. - Discuss this activity in their journal.

Prerequisite activities: Students will:

- Review related mathematical definitions and phrases: angles, tessellation, pattern, plane. - Review visual and algebraic patterns. - Explore tessellations using Geometer’s Sketchpad.

Extensions: See ‘Multiple intelligences used in this Lesson’ under ‘Adaptations for Diverse Learners.’ Using Performance Descriptors to Measure Student Achievement Distinguished The student demonstrates exceptional and exemplary performance with distinctive and sophisticated application of knowledge and skills that exceeds the standards in algebra. A student analyzes a given set of data for the existence of a pattern numerically, algebraically, and graphically. The student demonstrates exceptional and exemplary performance with distinctive and sophisticated application of knowledge and skills that exceeds the standard in geometry. A student develops the properties of tessellating figures and uses those properties to tessellate the plane.

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- Students will complete all tasks well and with little or no aid. - Patterns used to tessellate the plane will include at least two geometric figures. - All puzzle pieces are the same size and shape. - Connections between puzzle pieces are clear and on subject. - There is only one way to solve the puzzle. - The puzzle is highly rated by their peers.

Mastery The student demonstrates fundamental course or grade level knowledge and skills by showing consistent and accurate academic performance that meets the standard in algebra. Given a set of data, the student recognizes a pattern. The student demonstrates fundamental course or grade level knowledge and skills by showing consistent and accurate academic performance that meets the standard in geometry. A student develops the properties of tessellating figures and uses those properties to tessellate the plane.

- Students will complete tasks with some aid. - Patterns used to tessellate the plane will consist of one or more geometric figures. - All puzzle pieces are the same size and shape. - Most connections between puzzle pieces are clear and within the applicable subject categories. - There may be more than one way to put some of the pieces together. - The puzzle is moderately rated by their peers.

Novice The student demonstrates substantial need for the development of fundamental knowledge and skills characterized by fragmented and incomplete performance in algebra. Performance needs considerable development. The student recognizes visual patterns.

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The student demonstrates substantial need for the development of fundamental knowledge and skills characterized by fragmented and incomplete performance in geometry. Performance needs considerable development. The student recognizes some of the basic geometric figures.

- The student can only complete the task with aid or does not complete the task. - Patterns are very simple (e.g. squares) or may not tessellate the plane. - Puzzle pieces may be cut out in varying shapes and sizes. - Connections between puzzle pieces are not clear or not there. - The puzzle may not be able to be solved by most students. - The puzzle is rated poor by most of their peers.

Adaptations for Diverse Learners Check with the special educators for any modifications or considerations for students. Students will need to learn (or review) terms associated with this project. Multiple intelligences used in this lesson:

- Visual/Linguistic – Writing one or more paragraphs about tessellations using Microsoft Word and a data projector. This can be a group paper or an individual paper. Prompts can include a definition, where you can find tessellations, how they are made, and discussing famous artists who use them.

- Visual/Spatial – Drawing and coloring the tessellating puzzle; taking pictures of tessellations found at EFHS.

- Logical/Mathematical – Writing a mathematical rule within the puzzle; measuring and adding angle measures and researching tessellations on the web.

- Musical /Rhythmical – Writing/singing a song about tessellations. - Interpersonal/Social – Sharing, displaying and reconstructing the puzzles; designing tessellations in

a group. - Intrapersonal/Individual – Tessellating puzzle project designed by the student. - Bodily/Kinesthetic – Tessellating with body movements and formations. - Naturalistic – Tessellating with fall leaves, beehives, or honeycomb.

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Resources for Adaptations: Student Assistance Team Manual West Virginia Department of Education Division of Instructional and Student Services The Teacher’s Resource Guide Hawthorne Educational Services, Inc. 800 Gray Oak Drive Columbia, MO 65201 Telephone: (573) 874-1710 Fax: 1-800-442-9509 Math Madness Bob Garvey 1835 Princeton Dr. Louisville KY 40205 Including All Students Kansas Curriculum Center Kansas State Department of Education 120 E 10th Ave. Topeka, KS 66612 (785) 291-3097 Integrating Curricula with Multiple Intelligences (training manual) P153 IRI/Skylight Publishing Web sites: LDOnline

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Classroom Accommodation List Appropriate School-Based Accommodations and Interventions Integration of Technology Content Standard 3: Technology Productivity Tools Students will:

- Use technology tools to enhance learning, increase productivity, and promote creativity: and - Use productivity tools to collaborate in constructing technology-enhanced models, preparing

publications, and producing other creative works. Objectives:

- TEC.9-12.3.2 Select and use appropriate technology tool to efficiently collect, analyze, and display data that is relevant to class assignments.

The Geometer’s Sketchpad Key Curriculum Press 1150 65th St. Emeryville, CA 94608-9739 Instructional Materials Each student needs:

- Art supplies: o Tag board o Construction paper o Scissors o Markers and/or colored pencils o Tape or glue o Plastic bags

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- Tessellation handout - Directions for creating tessellations - Access to computer/internet/Geometer’s Sketchpad

Web-Based Resources Useful web sites: Mathforum.org

- Tools and lesson plans for prerequisite and/or similar activities such as investigating geometric figures and tessellations

NCTM.org

- The National Council of Teachers of Mathematics web site has dozens of resources. The Illuminations section is excellent.

Matti.usu.edu

- A web site devoted to virtual manipulatives, they have a section on creating tessellations.

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TESSELLATION PUZZLES

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TESSELLATION PUZZLES

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TESSELLATION PUZZLES

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DISTINGUISHED

WHAT ARE TESSELLATIONS ?

Tessellations are shapes that are put together and do not have gaps or overlaps. All shapes make patterns and are made with patterns. They could also be an optical illusion. The joining of angles add up to 360 degrees. The patterns repeat. The patterns all

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line up perfectly. You can use more than one shape to make a pattern. Circles do not tessellate. M. C. Escher made tessellations famous. His pictures are unique because he used tessellations. He lived in Italy and left in 1935. Some of his art was impossible. After he left Italy he went to Spain. He’s really made a lot of tessellations. They are fun to do and cool. The tessellations were easy and some were hard to do. Tessellations were time consuming to make. They are frustrating. Any idea could be used to make a tessellation. We all like to do tessellations. They can be colorful. There are 100 web sites on tessellations. Tessellations vary in sizes.

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MASTERY Tessellations

Tessellation is a way to tile a floor so there is no overlapping and no gaps. Tessellations come in many shapes and sizes. Tessellation means to form or arrange small squares in a checkered or mosaic pattern. Tessellation is to fit together in a pattern with no space in between. You can tessellate just about anything depending on what shape it is. They can be made into puzzles. You must have four adjacent angles in a tessellation and they must add up to 360 degrees.

M. C. Escher made many difficult tessellations. M. C. Escher was a famous artist for using tessellations. He was famous for creating spatial illusions and repeating geometric patterns. Escher’s work is found in many museums and art galleries around the world. Escher used a lot of animals, especially fish and birds.

Tessellations are fun and creative. It also takes a lot of patience. You

have to be able to think from the puzzle maker’s point of view. Making tessellations builds creativity. They can also be colorful. Tessellation can be difficult to draw. A basic tessellation consists of squares, triangles, and parallelograms.

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NOVICE

The World of Tessellations

Tessellations never overlap or have any gaps. They can go together like a puzzle. They have to cover the whole plane. If you are doing a single pattern, they are all the same size and shape. Each groove of the shape has to fit perfectly throughout the whole paper. Tessellations are rather fun to do. Some tessellations can be very confusing. M.C. Escher made tessellations famous by using animals and shapes. Some tessellations are very easy to draw. M.C. Escher’s tessellations are fun to look at. M.C. Escher got ideas for his tessellations when he traveled. Some of his works included optical illusions. He was one of the most popular artists who worked with tessellations. You can make pictures from of them. Tessellations are used in mosaics. Making tessellations can be frustrating.

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VISUAL-SPATIAL TESSELLATIONS

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TESSLLATION SONG

Its time to pattern the tessellation. We start with points, lines, and planes you see. And we build segment, rays and angles too. And polygons of any type that pleases you. Now we draw shapes of all sizes. Making repeating patterns that surprise you. So throw your text away and. Its time to avoid overlaps and gaps.

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With the same tessellation. *Partial credit to the song come from Math Madness by Bob Garvey. Additional words were added to fit the math topic.

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BODILY-KINESTHETIC TESSELLATIONS

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NATURE TESSELLATION