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Presentation transcript:

Tessellations Starting slide…. Needed - Computer with overhead computer projector, with power point software and internet….

What Is a Tessellation? A shape or tile that repeats to fill a surface without any gaps or overlaps. The name comes from the word tessella, the small square tile used in ancient Roman mosaics. Quick view of Core Curriculum Standards…. TIMMS article… note Standard 9 and Standard 13 of National Council of Teachers of Mathematics...NCTM NJ Core Curriculum Standards 4.7 K-4 - 2,3,4,5,6,7,10 8th - 12,13,15,16,19

What is a tessellation? Formal definition: A careful juxtaposition of elements into a coherent pattern sometimes called a mosaic or tiling. Simply put: A tessellation is like a puzzle that repeats a particular pattern. For example: Read…. To students…. Click anywhere when finished...

In nature and life, they appear often: Honeycombs... Mud flats... Read headings… Click when slide finishes... In games, like checkers..

Patterns Very basic tessellations are simply a repeating pattern. The rule is that you have to make sure that the shapes fit together leaving no open space on the page. Read to students… click when completed...

Professional Tessellation Designs Sun and Moon M.C. Escher Fish Robert Ingalls Many artists have created master works of art using the simple rules of tessellations. M.C. Escher and Robert Ingalls are among many tessellation artists. Can you you spot the repeating shape? Read to students… click when completed...

The mathematics of the Regular Tessellation... A regular polygon tessellation is constructed from regular polygons. Regular polygons have equal sides and equal angles. The regular polygons must fill the plane at each vertex, with repeating patterns and no overlapping pieces. Read to students… Note... links for student presentation.. Two links take you to internet... Polygon examples and Regular polygon examples Important concept… filling the vertex... click when completed... Note: this pentagon does not fit into the vertex… therefore it is not a regular tessellation.

A famous mathematician, Kepler studied tessellations and noted the regular tessellations or (tilings) of the plane. There are only three regular tessellations; one of triangles one of squares READ... Note… internet link ... to Kepler…. Click when finished…. and one of hexagons.

This is NOT a regular polygon tessellation, because... vertex The plane is not filled at the vertex, because there is space left over. space Read slide… Space at vertex should be noted. click when completed... A regular polygon tessellation, can be changed using “alterations” to the sides of the polygon. These alterations are called transformations.

Three Common Transformations 1. Translation, which is a slide of one side of the polygon. 2. Reflection, which is a flip or mirror image of one side of the polygon. 3. Rotation, which is a turn of a side around one vertex of the polygon. Read slide… There are other transformations … really combinations of these three… click when completed...

Translations - a slide this side the alteration moves here click when completed... Get practice template and try creating a translation….

Reflections - mirror images the alteration flips here Reflections….. Go through with audience…. Demo on board… check audience for correctness. Get practice template and try creating a reflection….

Rotations - turns the alteration here rotates around this vertex Go through with audience…. Demo on board… check audience for correctness. Please note: students have most difficulty here…with rotations… we limit rotations to 90 degree increments. Get practice template and try creating a rotation...

Extras ... Is this a regular tessellation? What kind of tessellation is this? Link to with the first two titles to have students complete “assessment”…. Is this a tessellation… What kind of tessellation is this… Java script internet site… great for whole class instruction… May need more class time to have students complete their own tessellations using a variety of software tittles.. Tessellation internet activity Cool site for interactive Javascript tessellations. Constructing tessellations - Claris/Appleworks, HyperCard, HyperStudio, Geometry Sketchpad, Windows Paint and more.

Links Totally Tessellated - ThinkQuest winner - great site, instruction, information. http://library.advanced.org/16661/ Tessellations Tutorials - Math Forum site http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html - site for construction of tessellations. http://forum.swarthmore.edu/sum95/suzanne/links.html - great list of tessellation links Math. Com - List of good tessellation links http://test.math.com/students/wonders/tessellations.html World of Escher site - commerical site with gallery of Dutch artist,Escher who was famous for his tessellation art. http://WorldOfEscher.com/gallery/ Science University’s Tilings Around Us Site. http://www.ScienceU.com/geometry/articles/tiling/tilings.html References…. Links… lots of good activities and materials available for additions activities. Link to end show… for slide show of tessellation…. Press escape to end slide show… and return to this page.. Other links from Forum. http://forum.swarthmore.edu/library/topics/transform_g/ The end ...