1. Pick’s Theorem By Harry Marshall

2. Summary Accompaniment to Glencoe’s Mathematics: Applications and Concepts Course 1 Chapter 14 – Geometry: Measuring Area and Volume; Lesson 14-1 – Area of Parallelograms, Lesson 14-2 Area of Triangles Chapter 9 – Algebra: Solving Equations; Lesson 9-5 – Solving Two-Step Equations 6 th Grade Mathematics Enrichment

3. Sunshine State Standards MA.B.1.3.1-1, MA.B.1.3.2-4, MA.B.1.3.3-2, MA.B.1.3.3-3, MA.B.2.3.2-2, MA.C.3.3.1-1, MA.D.2.3.1-2, MA.A.3.3.1-4, MA.A.3.3.2-2, MA.D.2.3.1-1, MA.D.2.3.1-3, MA.D.2.3.1-4 MA.D.2.3.2-1, MA.D.2.3.2-2

4. Prerequisite Skills and Supplies Basic Understanding of Computers Internet Access on a Computer with Java Installed Familiarity with Concept of Area Familiarity with Algebraic Equations

5. Objective To provide students with practice calculating area as well as introduce them to a new concept, Pick’s theorem without the need for the costly manipulatives, such as the geoboard and rubber bands customarily used to teach Pick’s Theorem.

6. Key Terms Area Equation

7. Count the number of grid points on the boundary of the polygon (b), and the number of grid points inside the polygon (i). The area (A) of the polygon is given by Pick's Theorem: A = i + b/2 −1

8. Questions 1) What is the area of the polygon (Use Pick's Theorem)? 2) What is the area of the polygon (Use Area= (1/2)BH)? 3) Do the answers match? 4) Drag the slider above to the right to check your answers. Now, move the slider back to the left and drag the red points to form a rectangle and answer the questions above. Remember, for a rectangle, A=LW. Now, try dragging the red points to form more complex polygons that you are able to find the areas for. Compare the area's you arrive at using the standard formulas for area with those you arrive at using Pick's Theorem. 5) How do the results compare? 6) Will Pick's Theorem work for any regular polygon constructed in a grid with points equal distances apart?

9. Key Questions Compare / Contrast Pick’s method for area with the formulas/methods you already know. Create three different polygons with the same area. What are the perimeters of each? Which of these has the largest perimeter (greatest number of boundary points)? Which of these has the smallest perimeter (least number of boundary points?

10. Assessment The answers to the questions are to be turned in and graded by the teacher and returned to the students with feedback. A participation grade can also be given for the class discussion.

11. Supplemental activities lesson Extensions to the lesson could include overlaying a grid onto a map to approximate land masses to show the students a way to estimate the areas of shapes too complex to get exact measurements for using formulas for area other than Pick’s Theorem.