Tag Archives: tessellations

Creating tesselations with 6th graders and Geometer’s Sketchpad


Katie Hildebrandt (6th grade math teacher) and I just finished a 2-day unit on Tessellations using Geometer’s Sketchpad to rotate an equilateral triangle and translate a parallelogram. I can’t praise Geometer’s Sketchpad (sometimes shortened to Geo Sketchpad, GSP, or simply Sketchpad) enough. It is one of the few pieces of educational software out there that is entirely constructivist. You can actually learn math by using GSP. Just like classical Euclidean Geomtetry breaks down everything in the world to points, lines, and planes, so does GSP; You can quickly learn to construct, animate, and measure a range of sketches, from the simple to the complex.

In this brief unit, we verbally discussed how a vector is a geometric object (in this case, a segment) that has both a direction and distance. They notice the “di” in the beginning of each word. I also tell them that translating is the same as sliding, and both words have an “sl”. We talked about angles of rotation, indications of symmetry, interior angles of a triangle, and reinforced vocabulary: equiangular, equilateral, congruent, parallel lines, etc.

I always start off by showing them works by M.C. Escher. On the site, there is a link to a gallery of his symmetry drawings. I marvel at how Escher painstakingly drew his incredibly intricate and fascinating tesselations component by component. I imagine his pile of pencil stubs and eraser shavings, and reinforce for the kids how we can create infinite variations with GSP in a matter of seconds by clicking and dragging.


I posted something about last year’s activity here.

There is a great PDF with multiple activities put out by Key Curriculum Press (@keypress) embedded below or you can click here to download it. We use pages 7-8 for the translation activity and pages 9-10 for the rotation activity.

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Tessellations with Geometer’s Sketchpad in 6th grade Math


I spent a few days this week with the 6th grade math teachers/students. At The School at Columbia University, 6th graders study Islam and Mecca as part of the grade-wide theme: How History Shapes my Identity. For the last few years, I’ve worked with the math teachers to show the kids how to design tessellations on the computer. Then, the students take their creations to Art class and build a physical model out of clay and/or paper. It is one of my favorite integrated projects. (All of our K-8 Themes and Concepts can be found here: http://theschool.columbia.edu/about/curriculum.)

Katie Hildebrandt, often confused from behind for one of her students (à la Macaulay Culkin), is an energetic and supportive member of the 6th grade team and has a natural gift for breaking down mathematical concepts for her students. Before Winter Break, we met and planned a 3-day mini unit for our first week back; It bridged a unit on solving equations with her next unit on the Cartesian Plane. We briefly went over how to make transformations and use specific menu options in Geometer’s Sketchpad, as Katie is one of those independent teachers that initially explores on her own rather than rely on my tutelage. Power to the people!

On the first day of the mini-unit, Katie led a class on reflecting a polygon over the x-axis and y-axis. Students explored the resulting coordinates and stated the formula as an algebraic expression. For example, reflecting over the x-axis means (x,y) becomes (x, -y).

The second day, Katie showed how to rotate a polygon 90, 180, 270, and 360 degrees around the origin. Again, students analyzed how rotating the figure affected the coordinates of the original shape. For example, rotating 90 degrees meant that (x,y) become (-y, x).

On the third day, I stepped in, and we talked about how a tessellation is a pattern of repeating shapes that do not overlap and have no gaps in between. I showed them some of M.C. Escher’s artwork, and we talked about how classic Islamic art would rely on geometric patterns rather than animal or human forms. Because of the nature of our curriculum, the students had similar discussions in Art and Spanish among other subject areas. I showed the kids how to use Geometer’s Sketchpad to build an equilateral triangle, alter one side, and rotate that side 60 degrees to create a new shape. Then we rotated this altered triangle 60 degrees 6 times to form a hexagon before we tessellated the whole hexagon.

For my tessellation activities, I use two online lesson plans that I located years ago:
http://ww3.wpunj.edu/icip/itm/Lessonpl/sketch/rotate.htm – Triangle Rotations by Janet Mae Zahumeny of Roselle Park High School
http://mathforum.org/sum95/suzanne/tess.gsp.tutorial.html – Parallelogram Translations by Cathi Sanders of Punahou School

I started exploring/playing with Geometer’s Sketchpad in 1994 as an undergrad at Bryn Mawr College. To this day, it remains my favorite piece of educational software. Not too long after, I learned about The Math Forum – an amazing resource for math teachers and students founded at Swarthmore College and now seems to be hosted by Drexel University. There is a great link about Exploring and Creating Tessellations: http://www.teacherlink.org/content/math/activities/skpv4-tessellation/home.html

Geometer’s Sketchpad resources:
General Resources: http://www.dynamicgeometry.com/General_Resources.html
Resource Center: http://www.dynamicgeometry.com/
Sketch Exchange: http://sketchexchange.keypress.com/
Workshop Guide: http://www.dynamicgeometry.com/Instructor_Resources/Workshop_Guide.html


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