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Real World Math: Concept Lessons - Tessellations w. SketchUp

Tessellation is a math topic that is usually found off the beaten path but they contain important geometry and measurement concepts.

A simple tessellation can be seen in a tiled floor but one full of transformations can truly be a work of art.  This activity takes students through the steps of using three different types of trans-formations to construct tessellated patterns.  You don’t need to be a strong math student or artistic to create an interesting design.

SketchUp is a free 3-D modeling application from Google, but it can also be applied for two-dimensional drawings as well.  There are numerous applications available for students to construct tessellations with, but using SketchUp to perform the task has several advantages.  First, this is a good introduction to the SketchUp application.  Only a limited use of the full tool kit is necessary.  When students become comfortable with the program then they will be ready for bigger and better things.  Additionally, some of the other tessellation applications do all of the work for the user.  One only needs to grab a point and drag and the transformation is effected.  With SketchUp, the mathematical concepts are recognized and applied.  The Rotation needs to be rotated by measured degrees and so on.  And finally, SketchUp can be used to illustrate how area and perimeter is preserved in the shapes.

The main download attached is a tutorial on how to perform several transformations in SketchUp.  Slide, rotate around a midpoint, and rotate around a vertex transformations are demonstrated in a series of SketchUp scenes.  The file could be used an instructional tool or for self-directed learning.  I’ve also included some tessellation samples and some blank versions for younger grades to color.  Additionally, a file that contains a cube with a tessellated pattern is available as a download.  This is a Four Color Map problem, that is – the cube can be colored in such a way that only four colors are needed and no colored region shares a border with the same color.  See the Four Color Map Problems lesson on the Exploratory Lesson’s page for similar exercises.  Even with borrowing ideas from Escher, this was extremely difficult to make.  As yet, no tutorial has been made for constructing a tessellated polyhedron.

As previously mentioned, a minimal amount of SketchUp tools is needed to complete the activity; however, you should become well acquainted with the program before using it for instruction.  The main tools used to construct the tessellations are as follows:  Ruler, Select, Move, Draw, Polygon, Protractor, and Paint Bucket tools, and Copy and Paste commands.  Orbit, Pan and Zoom tools are used for changing viewing angles.

A free version of SketchUp is available for download from Trimble @ https://scitechinstitute.orgsketchup.google.com/download.html.  Once students have the program, they can use SketchUp without an Internet connection but the most useful help guides are online.

Objectives

• Construct tessellated patterns using SketchUp
• Demonstrate translations: slide, rotation around a midpoint, and rotation around a vertex
• Explore and apply combinations of transformation and color patterns using SketchUp’s tools

• Math Is Fun website – Link
• The Official M.C. Escher Website – Link

Technology Topics
3-D Modeling, Online & Software Tools
Math Topics
Geometry
K-6, Middle School

Real World Math

Lesson Plan