Geometric Nets: Feeling Flat?

Lesson Summary: This lesson is designed to build on geometric concepts by using nets to represent 3- dimensional figures and to deconstruct those figures back into nets. This will help students understand the concepts behind calculating total surface area and will also aid students in being able to recognize different nets for the same 3-dimensional figure.

Subject: Math: Geometry

Grade Level: Target Grade: 7 Upper Bound: 8 Lower Bound: 7

Time Required: Introducing the worksheet should take about 10 minutes. The activity will take about 30 minutes.

Authors: Graduate Fellow Name: Bruce Ngo Teacher Mentor Name: Elaine Stallings Date Submitted: 2/27/2006 Date Last Edited:

Required Materials: Polyhedron Assembly Kit (or reasonable substitute) Worksheet with Nets Rulers (for worksheet)

Lesson Introduction / Motivation: How do we represent 3-dimensional figures in two dimensions? What is a geometric net? Students will be encouraged to design different nets for the same 3-dimensional figures and use them to get a better understanding of surface area of a solid. Students will construct polyhedra and compete for the “best” net. Lesson Plan: 1. The students should go over the worksheet quickly prior to the activity. The teacher should discuss the directions with them and make sure that they understand the concepts of area, faces of a polyhedron, and that total surface area is the sum of the areas of each individual face. They should also be able to calculate the area of each face by measuring the edges and using the appropriate formula. Conceptually, the worksheet progresses in difficulty. The first figure represents a cube, which is a very common net that the students should be familiar with. The second net represents a cylinder, which students may find slightly harder to recognize but the net can be used to demonstrate the concepts behind calculating lateral surface area. 2. The students are free to cut out and assemble the paper models on their own time or in class if time permits. This may be done as a separate activity before or after the net activity. Coloring the paper nets may help to reinforce the concept of surface area. 3. For the activity, each group should have access to the basic pieces for construction of two different simple polyhedra. For example, the groups received 6 interlocking squares to build a cube and 8 interlocking equilateral triangles to construct an octahedron. In this case, the students may not be familiar with an octahedron but they should be able to deduce that it has eight sides. Preassembled figures may be useful to provide examples of the figures to build. 4. Once each group has assembled the polyhedra, they should open them up in order to design the net for the polyhedron. The students should be encouraged to design interesting and “original” nets as the instructor will choose the best net from each polyhedron. As an alternative, the class may vote on the most interesting net. 5. The winning groups should be invited to the front of the class, where they will construct a fairly complex polyhedron and then design a net for it. For example, they may be invited to construct a dodecahedron using 12 regular pentagons. The other groups can be allowed to construct various other figures using their pieces or work on the worksheet. 6. Once the two groups complete this task, the class should vote on the most interesting net. The two groups may then be rewarded, with the winners receiving first prize.

Lesson Closure: The students should finish the worksheet if they have not done so already. Encourage the students to identify other polyhedra they may observe and design nets for them. They can also visit http://www.korthalsaltes.com/ for printable paper cutouts for all sorts of polyhedra.

Learning Objectives: The student should be able to calculate area of a shape and total surface area of a 3- dimensional figure. The student should be able to identify the 3-dimensional figure from its net.

TEKS covered: 7.8 (B) Make a net model of the surface area of a solid. 8.8 (A) Find the surface area of prisms and cylinders using models and nets.

Prerequisites for this Activity: Students should be familiar with what a net is and what it represents. They should also know the names of several simple 3-dimensional figures. They should have formulas for calculating the area of triangles, circles, squares, and rectangles.

Background & Concepts for Teachers: From http://en.wikipedia.org/wiki/Net_(polyhedron): In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for models of polyhedra to be constructed from material such as thin cardboard.

References:  Images of nets for the lesson plan were created by George Olshevsky ©2004, and can be found at http://members.aol.com/Polycell/nets.html.  Images of nets for the worksheet were obtained at http://intranet.cps.k12.il.us/ from the Day 155 to 156 Structured Lesson Plan for High School Geometry.  Paper model cutouts can be found at http://www.korthalsaltes.com/.  More information on nets and related subjects can be found at http://en.wikipedia.org/wiki/Net_(polyhedron).  Examples of polyhedron construction kits include ones that can be purchased at websites such as http://www.polydron.co.uk or http://www.jovotoys.com/. Possible substitutes include using toothpicks and gumdrops or taping cardboard cutouts of regular polygons together to construct the polyhedra.