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Project Summary

Difficulty	5
Time required	Short (several days)
Prerequisites	None
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsor

Sponsored by a generous grant from Motorola

Abstract

Objective

The objective of this project is to study lattice polygons and prove that Pick's Theorem is correct.

Introduction

The strange shape below is an example of a lattice polygon, which is a polygon whose vertices lie on points in the plane that have integral coordinates.

As you can see, it is a complex shape, but there is an easy way to calculate its area, by simply counting lattice points! If you count the number of lattice points on the boundary of the polygon (b), and the number of lattice points inside the polygon (i), then the area (A) of the polygon is given by Pick's Theorem: A = i + b/2 −1.

A good way to explore lattice polygons is with a geoboard. A physical geoboard is a piece of wood with pegs (or nails) arranged in a regular grid. The wood represents a section of the plane, and the pegs or nails are the lattice points. You stretch rubber bands over the lattice points to create polygons. You can make or buy a geoboard for this project (it would make a nice addition to your display), but it's not absolutely necessary. See the Bibliography for an online geoboard program.

Terms, Concepts and Questions to Start Background Research

To do a project on Pick's Theorem, you should do research that enables you to understand the following terms and concepts:

• polygon
• lattice polygon
• area of a rectangle
• area of a triangle
• geoboard

Bibliography

• The National Library of Virtual Manipulatives for Interactive Mathematics (that's a mouthful—must have a really big sign), has lots of cool programs that give visual insight into math concepts. Here is a link to their online geoboard program (Java applet) which you can use in your investigation of Pick's Theorem:
  http://nlvm.usu.edu/en/nav/frames_asid_282_g_3_t_3.html?open=instructions
• This NLVM link has a big selection of cool programs with Geometry themes for 6th–8th grade students:
  http://nlvm.usu.edu/en/nav/category_g_3_t_3.html
• This webpage has some information on Pick's Theorem, and a hint for how to go about proving it:
  http://www.math.hmc.edu/funfacts/ffiles/10002.2.shtml

Materials and Equipment

• A computer with Internet access and browser with Java Runtime Environment installed is needed for the geoboard program.
• You can make your own geoboard with a piece of wood, some nails and rubber bands. You'll want at least a 5 × 5 grid of nails. Make your grid spacing comfortable for your fingers (and not too big for your rubber bands).

Experimental Procedure

1. The online geoboard software includes a complete set of instructions. You can use as many rubber bands as you want. You can select and color any of the polygons you construct. When you click on the "Measure" button, the program will show you the perimeter and area of the currently selected polygon.
2. First, try out Pick's Theorem on some polygons you create on the geoboard. Start simple and get more and more complex. Predict the area according to the formula, then click the "Measure" button and see if you were right.
3. For practice, try to create a polygon that looks like this one:
   
4. Now all that's left to do is to prove Pick's Theorem. Divide and conquer!

Variations

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas.

Credits

Last edit date: 2006-02-22 00:22:03

Career Focus

If you like this project, you might enjoy exploring careers in Pure Mathematics.

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