Tuesday, April 22, 2014

Geometry Tech Tool: Geometer's Sketchpad Tessellations that use Rotations

You'll see in another post that the Geometer's Sketchpad and it's iPad version: Sketchpad Explorer are wonderful tools to use to complement a PBL/PrBL classroom. Below is a review of a specific tech tool that uses the software, written by a colleague of mine: Terry C.

Original CEP Posting


Curator: Terry C.

Name & Link to Tech Tool or Resource homepage: Tessellations That Use Rotations

Link to parent Wiki page: Geometer's Sketchpad

Link to Other GSP Activities: Geometer's Sketchpad Activities

Brief Description of Specific Math Activity: Students construct an irregularly shaped tile based on an equilateral triangle, and then use rotation to tessellate the plane with it (Key Curriculum Press, 2009).


Description of Learning Activity

By creating the tessellation and then dynamically changing the original tile to see the effects, students get a deeper understanding of what makes the tessellation work (Key Curriculum Press, 2009).

1. Learning Activity Types

  • LA-Present - (read or attend to) presentation of new content/ideas
    • LA-Present-Demo - demonstration
    • LA-Present-Explain - explanation
  • LA-Explore - exploring/investigating mathematical ideas
  • LA-Apply - applying mathematics to problems and situations

2. What mathematics is being learned?

The objective of this activity is to use rotation to tessellate and to explore rotational symmetry (Key Curriculum Press, 2009).

NCTM Standards

  • NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
  • NCTM-Geo-specify locations - specify locations and describe spatial relationships using coordinate geometry and other representational systems;
  • NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

  • PS-conceptual understanding - Students work with a specific example but the activity tries to promote the idea that students can wander down a path of "What if I did this?" and still reach similar results.
  • PS-adaptive reasoning - Throughout the construction and manipulation, students have to understand what is happening, how its happening, and why its happening.
  • PS-productive disposition - Seeing the tessellation animated at the end of the activity is fun and rewarding.

Prerequisite Knowledge Required: Experience with equilateral triangles, translation, tessellation, and rotation.

3. How is the mathematics represented?

Mathematics can be represented symbolically and graphically in this activity. Geometer's Sketchpad allows for both dynamic and static representations of mathematics, and it depends on the inputs made by the creator how the mathematics is represented in the tool. Through the instructions, this activity guides students to creating graphical and symbolic representations of mathematics that act as virtual manipulatives.

4. What role does technology play?

Geometer's Sketchpad makes a unique contribution in that it automates tessellation of a plane through tile rotation, so that students can see the effects more easily and quickly.

Affordances of Technology for Supporting Learning

  • Computing & Automating - Sketchpad allows students to construct shapes on a computer more easily than if they were drawing them with pencil and paper. Sketchpad also allows them to manipulate and change the shapes more easily than with pencil, paper, and eraser. The effect of rotating the tile to tessellate the plane is achieved through technology; this cannot be done on paper.
  • Representing Ideas & Thinking - Sketchpad allows students to visualize and explore mathematics problems.
  • Accessing Information -
  • Communicating & Collaborating -
  • Capturing & Creating -

5. How does the technology fit or interact with the social context of learning?

According to the instructions, this activity is meant to be performed in student pairs. It can also be modified for a whole class. The technology facilitates the ease in which students can construct tessellations together (or side-by-side on separate computers) and then share the results with each other. Using technology allows students to rotate the original tile dynamically, which certainly is not easily accomplished with pencil and paper.

6. What do teachers and learners need to know?

This activity requires that students and teachers have access to Geometer's Sketchpad, and intermediate knowledge of Sketchpad is required to do this activity. The suggested grade level is 9 to 10; however, advanced seventh or eighth grade students could work on this activity. The activity includes an introduction on the activity for the teacher, notes, instructions, and discussion points for the teacher to guide the class through the activity, and a worksheet for the students that will guide them through the activity in Sketchpad and ask them questions at various intervals. The recommended duration for this activity is 45 minutes.

How this Supports and Supplements PBL/PrBL
 This activity is a great way for students to work on developing a conceptual understanding of tessellations as well as the procedural process of making tessellations, both of which can be daunting and time consuming for students. A lot of great projects and problems can focus on tessellations, or can take a tessellation turn (try turning ANY tiling problem into a tessellation problem!). The exposure students get by using this tool and doing this activity helps familiarize them with the concepts well enough to prepare them for thinking this way in projects and problems.