NETWORKS: FROM POINT TO POLYHEDRON

Bibliographic Information:
Thomas, Christine D. and Carmelita Santiago. (2002). “Building Mathematically Powerful Students Through Connections.” Mathematics Teaching in the Middle School, May 2002. Vol. 7, Iss. 9, p. 484-488.

Mathematical Concept: Exploring Networks with Polyhedron

Grade Levels: 6th – 8th

NCTM Principles and Standards for School Mathematics:
• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Materials:
75 colored non-bendable straws
Yarn, twine, dental floss, or string
Scissors
Rulers
Protractors
Bamboo Skewers
Sharp Knife
Instruction Sheet

Student Materials:
Notebook/Portfolio for Research
Calculator
Colored Pencils/Markers

Detailed Description:
This activity is based on parts from the “Building Mathematically Powerful Students through Connections” article. Prior to completing this unit, the instructor is encouraged to design each of the straw polyhedron and display them around the room. The models will be used to investigate concepts such as: point line, line segment, rays, angles, and networks. While engaged in this project, students will view themselves as mathematicians and researchers. As models are developed by the students, they will explore two-dimensional figures through investigations of polygons and their properties. Finally, the teams will delve into three-dimensional polyhedrons. Students will examine their string of triangles in terms of network, which is a figure made up of points (vertices) connected by non-intersecting edges (arcs).

Procedure:
1. Organize students into research teams of 2-4 students
a. Each student’s mission will be to contribute to the team’s development of the model, acquire an understanding of geometry concepts and relationships among the concepts, extend the team’s thinking about mathematics through research, and share and discuss insights with the team.
i. Through investigations, students should be able to find congruent angles: alternate interior angles, alternate exterior angles, corresponding angles, and supplementary angles.
b. The students will maintain a research portfolio in which they will keep:
i. A log of daily investigations, facts, and findings
ii. Charts, diagrams, and classroom notes to support investigations, facts, and findings
iii. Make inferences or speculations based on conclusions from the investigations
iv. A vocabulary list of new terms
v. Written reflections for each day’s activities

2. Students will be assigned to find potential connections to real-life situations through research, exploration, and mathematical discoveries.
a. Each team will have a person to work on the following tasks, or the teacher may divide each group of 4 as one specific team (i.e. Group 1 will be the Research Team, etc.)
i. The research team will investigate how the structures changes from adding new straws and/or vertices, analyze the properties of triangles, parallelograms, and trapezoids, and include the study of parallel lines .
ii. The measurement team will find the following measurements: the length of the straw, angle measures, and the area formulas for the triangle, trapezoid, and parallelogram.
iii. The algebraic team will create a chart and look for patterns to determine the number of straws required for a string of equilateral triangles of any length, as well as determine a formula for finding the number of straws to create a string of n triangles. (2n + 1)
iv. The discrete mathematics team will use the horizontal string of triangles to define and investigate networks. This group will also conduct internet research on mathematician Leonhard Euler, specifically looking for information on his work in graph theory.

3. Network: After gaining some background knowledge of the study of networks, the students will examine their string of triangles in terms of networks.
i. Students will examine the number of edges leading into each vertex and number each vertex accordingly.
ii. Discussion should ensue about finding an Euler path (traversing all edges without using any edge twice)
iii. Students should discover that they must begin at a certain vertex to completely travel the network and pass through every edge once and only once.
iv. Euler’s Theorem: If a network has 2 or fewer odd vertices, then it has at least 1 Euler path.
v. Students should conclude that because the string of connected triangles was a network with 2 odd vertices, they must begin and end at an odd vertex to trace an Euler path.

4. Pass out the instruction sheet titled Straw Polyhedra & Other Nets

5. Pass out copies of Polyhedra outlines as accessed on “Let’s Face It” (from PBS.org Mathline) for students to use as visual guides while constructing the Polyhedron networks. Students can use these handouts as guides for what the polyhedra should look like when it is lying flat.

Attachment: Worksheet/Activity Student Page
• Instructions as taken from the following website:
http://www.math.nmsu.edu/~breakingaway/Lessons/straw/straw.html

Instructions

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