Home > Developing Mathematics for Modeling > Polyhedra Investigations
   
   Polyhedra  
    Introduction
 

Introducing Polyhedra (Polyhedra are 3-d solids composed of polygons) aims to introduce students to mathematical exploration and investigation, and to the important roles played in mathematics by classification and definition. By defining the characteristics of categories and testing their definitions, students explore the structure of space-What's an edge? Vertex? Face? Side? Students build and explore qualities of 3-d solids, develop classifications of their constructions, and come to see the value of clarifying terms and definitions for communication and understanding. The lesson culminates with exploration of the Euler characteristic (F + V - E = 2), a pattern that unifies all polyhedra, no matter how different they appear.

 

Big ideas / Rationale / Questions to address / How to model student outcomes

 
    Big ideas
 

  • Constructing: Each student creates several different structures using Polydrons. There are no restrictions on what can be constructed, other than availability of sufficient material.

  • Recording: Each student records each construction in a math journal, including drawings and text. The text should describe the constructions in a way that differentiates among them. Each construction is labeled.

  • Traveling: Students display their constructions, so that they are visible to everyone.

  • Classifying: One student places her construction before the group. She tells its name and describes it. A list of descriptors is begun. Then, three instructional variations are available:

    1. The teacher selects a second student and asks that the student bring her construction and decide whether or not the construction is similar to (same pile) or different from the first (new pile) - and why. The list of descriptors grows as this process is repeated. Although after the first two, it is modified for efficiency - perhaps students bring all their constructions.
    2. The teacher asks the class for another construction that is "like" the one on display. One is selected and placed with the first. The class discusses what akes the two constructions alike. Any differences are also noted. This process continues until all the "alikes" are exhausted. The teacher asks for a construction that is different from the first group.
    3. The work is done in small groups of 4, and students present the results of their classifications and rationales.

The teacher's role is to help students see the value of clear expression-description leading to classification and names that inform. The teacher asks whether new descriptors are really new or are just re-expressions of the same ideas. The conversation is steered to include words describing faces, edges and vertices.

  • Finding relationships: Working in small groups, students attempt to find relationships among the number of edges, faces, and vertices of polyhedra. In whole group discussion, teachers help students to see the value of tables as a way to coordinate descriptions and the value of symbols as compact ways to express relationships. Why does the relationship work? Which expressions are equivalent? Why? (e.g., F + V - E = 2, F + V = E + 2)
  • Testing the limits: Conjecture. What happens to relationships among faces, edges, and vertices if one face is removed from a polyhedron?
    Rationale (Why this task?)
 
  • Mathematics is more than number. We begin outside of number so that preconceived notions of competence, talent, ability are avoided. Everyone begins in approximately the same place, because very little geometry is studied in elementary school and, even if taught in grades 1-4, it doesn't look like this.
  • We begin instruction with a strand of math that teachers explored in workshops this summer. We encourage teachers to apply what they learned and extend their thinking about geometry by working with students on investigations that teachers conducted at professional development meetings.
  • Provide teachers and students with experience in the investigative aspect of mathematics: Students explore, construct, discover, conjecture, and develop rules (characteristics that allow for classification/definitions). They test, compare and contrast, confirm/revise, retest, and re-revise.
  • Students discover the importance of communication in math and the need for precision in language by working with/without others to describe and define what they are seeing/thinking.
  • Students develop equivalencies and equivalence classes with the polyhedra. They experience the role of general criteria in constructing equivalence.
  • Students develop experience with different representational media and formats including Polydron models, drawings, text, tables, graphs, charts (visual representation), and notational symbol systems (e.g, Euler characteristic equations).
  • Students are supported to consider how mathematics is a search for pattern via the Euler characteristic.
  • Students are encouraged to explore the range covered by the Euler characteristic for polyhedra. What happens when the solid has one hole? Definition as a form of thinking.
    Questions to address    
 

  • What is the purpose of rules (characteristics)? [Note: Rules is a term from natural language familiar to students. More conventionally, we mean attributes or characteristics of a figure that define it. Later in the lesson, rules are stretched to encompass equations, and this will later be the dominant use of "rule."]

  • What is a definition?

  • How do you tell someone else about the characteristics? [e.g., How do you know that how you are thinking about an edge is the same way someone else is thinking about it?]

  • How do you know when you have enough information to conclude that your rule (chracteristic) is accurate?

  • What happens when something doesn't seem to fit?

  • How do we express a rule?

  • What are the functions of definition? Why does it matter?

    Model target student outcomes by
 
  • Drawing a pencil or block on the overhead
  • Writing characteristics of object in notebook
  • Writing definition of object
  • Testing the robustness of the definition with other objects (obvious examples and non-examples)
  • Revising written work and drawings based on above
Last Updated: July 11, 2005
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