Investigating Pick's Theorem

Many students may have seen Pick’s Theorem in middle school. In this set of lessons, students rediscover the theorem, use algebra to determine the coefficients of the equation, and explore the concept of change as a mechanism for finding the coefficients of Pick’s Theorem.

Math Content

• Systems of Equations
• Rates of Change

Although no single lesson in this unit addresses connections and representation by itself, the entire unit focuses on the Connections and Representation Standards by allowing students to make connections among mathematical ideas and asking students to use various representations to organize their work.

Individual Lessons

Lesson 1 - Rediscovering the Patterns in Pick’s Theorem

Use a geoboard, geoboard applet, or Geometer’s Sketchpad^® to help students discover the pattern of Pick’s Theorem.

Lesson 2 - Pick’s Theorem as a System of Equations

Students will gather three examples from a Geoboard or other representation to generate a system of equations. The solution will provide the coefficients for Pick’s Theorem.

Lesson 3 - Rates of Change in Pick’s Theorem

Students will use a spreadsheet to investigate rates of change among various figures created on a geoboard. The coefficients of Pick’s Theorem are easily determined from these rates of change.