Megan Urbanik

Math 5010

Website Activity

Title: Regular Polygons that Tessellate/ Irregular Polygons that Tessellate

Author: Megan Urbanik

Topics: What shapes tessellate and why. 

Connection to Core Curriculum:

 CCSS.Math.Content.HSG-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

CCSS.Math.Content.HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

CCSS.Math.Content.HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CCSS.Math.Content.HSG-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Overview: This activity will help students understand what a tessellation is and some of the basic mathematics behind it.  It will allow them to play with and discover some of the types of tessellations that can be done and begin to understand why they work.  

Objectives: Students will use tangrams to discover what regular polygons will tessellate and create a conjecture as to why certain shapes will tessellate.  They will also explore creating tessellations using paper and drawing one as well as using online resources to create them.  

Materials Needed: Tangrams, paper, polygon paper cutouts, scissors, tape, crayons, worksheet, and computers.  

Web Reference: http://gwydir.demon.co.uk/jo/tess/tess.htm

                    http://www.winslow-es.u52.k12.me.us/tc/tc.html

                    http://www.shodor.org/interactivate/activities/Tessellate/

http://www.pbs.org/parents/education/math/games/first-second-grade/tessellation/

Activity Plan: Students will be provided graph paper, rulers, protractors, and tangrams on their tables.  They will also be given a worksheet and I will explain the direction on it.  I will also refer them to the word wall as a reminder to some of the term they will see in tessellations such as symmetry, reflection, rotation, translation, etc.  Students will use the tangrams and protractor to discover why some regular polygons will tessellate and others will not.  The will be lead to understand that at any given intersection/point of tessellation the degrees must equal 360 degrees.  They will also be lead to discover that polygons that tessellate don’t have to be regular to tessellate.   Then they will be able create their own tessellation using whichever polygon they wish and whatever modifications as well.  After creating their own tessellations, students will use the links above (same as the tessellation creator links on my website) to create more and different types of tessellations.  These sites will help them to understand how shapes can be modified differently and thus tessellated differently (translated, reflected, rotated, etc.).   

Background: I wanted to do this activity because I made tessellations in grade school but I never really understood why.  I felt like it was art class rather than mathematics class.  We never went into any of the geometry behind tessellations, we were just given a square and told to cut out one side and tape it to the other then draw it over and over again until the paper was covered.  I wanted to show students that there is a reason to this and help them understand some mathematical terms that are used when doing tessellations. 

Included documents: Tessellation worksheet

References:  http://gwydir.demon.co.uk/jo/tess/tess.htm

http://www.winslow-es.u52.k12.me.us/tc/tc.html

http://www.shodor.org/interactivate/activities/Tessellate/

http://www.pbs.org/parents/education/math/games/first-second-grade/tessellation/

http://mathforum.org/sum95/suzanne/whattess.html

http://www.tessellations.org/methods-diy-papercut.shtml#

Activity Plan Worksheet

Tessellations

NAME:______________________

From looking at the two geogebra applets and pictures shown on the board give your own definition of a tessellation?

Tessellations are made from translating simple shapes without any gaps or holes.  Use the material provided on your table to decide which regular polygons will tessellate and produce a conjecture as to why they tessellate.  

Write in the box below your conjecture as to why certain regular polygons can tessellate:

When you are finished with your conjecture, use one of the paper polygons and blank paper on your table to create your own tessellation.  Use any of the processes discussed to do this.  

Teacher Reference:

At any given point the degrees must be equal to 360 degrees.  Thus the only regular polygons that can be tessellated are triangles, squares, rectangles, and hexagons.  Not all polygons have to be regular to be tessellated.  All triangles and rectangles can be tessellated, but hexagons have three special cases that will work for irregular polygons.