Geometry, Measurement, and Geofix

by Don S. Balka, Ph.D.

The study of two- and three-dimensional geometry is much more exciting and meaningful when students use manipulatives rather than attempting to learn about shapes from pictures on textbook pages. In particular, the action verbs sort, analyze, construct, classify, identify, measure, and explore generate added enthusiasm in classrooms when colorful materials are available for students to use.

All of these terms are major components of the NCTM Principles and Standards for School Mathematics (2000) and the Curriculum Focal Points (2008). Additionally, in the new NCTM document Focus in High School Mathematics (2008), conjecturing about geometric objects is cited as one of the key elements in reasoning and sense making with geometry.

Geofix, Polydron, and other interlocking shapes are excellent tools for providing students with opportunities to make their thinking about geometry visible to themselves and their teachers. Activities with these interlocking shapes help students move from concrete to abstract ideas about geometry and measurement.

A Prime Tool for Implementing Math Standards
Beginning with the Grade 3 Common Core State Standards for geometry and measurement, along with current standards from a number of states, we find many commonalities that lend themselves to the use of Geofix and other interlocking shapes. These include:

• Classify two-dimensional shapes based on the number of edges (sides).


• Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines.


• Create and describe triangles, rectangles, parallelograms, trapezoids, and other polygons.


• Measure the distance around objects.


• Determine the area of a two-dimensional shape.


• Classify triangles.


• Classify angles as acute, right, or obtuse.


• Apply translations, reflections, or rotations to figures.


• Create and classify three-dimensional figures, including prisms and pyramids.


• Classify three-dimensional figures by the number of vertices, faces, and edges.


• Create a net for a three-dimensional figure.


• Identify lines of symmetry for two-dimensional figures.


• Locate points in a coordinate plane.


• Explore the result for area or volume when the length of a side of a square or cube is doubled.


• Find the surface area of rectangular prisms.

Understanding Concepts Through Hands-On Discovery
Vertices, faces, and edges are common terms associated with the study of geometric shapes, yet students collect misconceptions when they are unable to visually see and physically manipulate shapes. Identifying figures by name while simultaneously gathering hands-on data help students to make conjectures that lead to important generalizations. For example, exploring vertices with Geofix could lead to discoveries such as the following:

• For prisms, the number of vertices is twice the "Name": Triangular prism (2 x 3 = 6 vertices), square prism or cube (2 x 4 = 8), pentagonal prism (2 x 5 = 10).


• Pyramids, however, provide a different relationship; the number of vertices is "Name plus 1": Triangular pyramid (3 + 1 = 4 vertices), square pyramid (4 + 1 = 5), pentagonal pyramid (5 + 1 = 6).


• Vertices of figures can be classified as odd or even, depending on the degree of (number of edges meeting at) a vertex. If we consider the sum S of the degrees for a figure, what conjectures can we make about this number? A cube, for example, has 8 vertices, each with degree 3. So, the sum S of the degrees is 24. For a triangular prism, S = 18.

Many state assessments include items involving nets. Using the "hinged" 3-D figures they create with Geofix, students are able to quickly unfold a three-dimensional figure to produce a two-dimensional net, fold it back together, and then unfold a different net. Small group settings provide opportunities to explore, for example, all the possible nets for a cube. A related activity would be providing sketches of nets to students and having them use Geofix shapes to create the three-dimensional figure. Classroom experience with nets has shown that students soon can identify and describe figures without using the interlocking shapes.

Line symmetry activities in which one student creates a figure next to a given line of symmetry and a second student constructs the mirror image of the figure allow students to quickly check their image by folding one figure across the other to see if they match.

Geofix and Science Class
Making connections between geometry and the real world, in particular science, is also a critical component of our classroom work. Many minerals and gems have crystals with specific three-dimensional shapes that can be modeled with Geofix. For example, galena and halite crystals occur as cubes. Crystal systems or categories established by scientists describe the three-dimensional shapes formed and the symmetries involved.

Soapy water and Geofix shapes combine to illustrate the physics of liquid surface tension. Although the mathematics is formidable and well beyond the bounds for intermediate and middle school students, excitement, surprise, and amazement will reign when students dunk a triangular pyramid into a soapy mixture and observe the resulting bubble with a special interior point (Steiner Point). The explorations continue when a cube is dipped into the mixture. First, a centrally located square appears as part of the bubble. More excitement follows as students do quick multiple dips of the cube, and a cube within the cube (hypercube or tesseract) appears.

The "Doing" is Key
As noted in the NCTM Standards, learning geometry requires both thinking and doing. Geofix and other commercially available interlocking shapes are the tools that make it possible for students to perform the doing. When these versatile manipulatives are a regular feature of the math classroom, visible thinking and excitement about geometry concepts follow.

Geofix Downloadable Activity Pages

Snapping together the brightly colored plastic Geofix shapes is a satisfying, hands-on way to explore geometry concepts such as perimeter and area, line symmetry, surface area and volume, and nets.

Click below for a ready-to-use sample activity from 2D and 3D Geometry with Interlocking Shapes by Don Balka and Mary Porter. This new Didax book features 15 exciting reproducible activities for introducing geometry concepts to students in grades 3-5.

Sample Geofix lesson: Net to Net

Geofix Internet Links

The more students play and learn with Geofix, the more they realize how amazingly versatile these manipulatives really are. Challenge your students to build a giant octahedron or towering helix, or to learn all they can about cubes and nets. The links below will lead you to some exciting Geofix resources on the Web.

	Didax.com: Geofix Models Gallery
	NCTM.org: Building a Box Activity

November Newsletter Theme: Interactive Whiteboards

Next month Class Ideas looks at a technology that's making an appearance in more and more classrooms: the interactive whiteboard. We hope you'll join us in November as we examine the benefits and drawbacks of this dynamic classroom tool, offer ideas on how to get the most out of your IWB, and bring you a fantastic special on Didax's new products for the whiteboard.