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Rediscovering the Patterns in Pick’s Theorem

9-12
1
AlgebraMeasurement
R. Alan Russell
Location: unknown

Students will use a geoboard, geoboard interactive, or Geometer’s Sketchpad® to help them discover the pattern of Pick’s Theorem.

The first lesson of this unit serves as an introduction to Pick’s theorem for students who have never encountered the theorem, as well as a refresher to those who may have forgotten the results.

Give students some version of a geoboard for investigation—you can provide them with a physical geoboard and rubber bands; you can distribute geoboard paper; or you can have them use the E‑Example Geoboard Applet, if computers are available. Ask them to create several figures using rubber bands noting the number of perimeter pins (P) of each figure as well as the number of interior pins (I). Ask students to also make a note of the area (A) of each figure. You may want to encourage students to create a chart to record their findings:

pdficon Geoboard Paper 

appicon Geoboard Interactive 

Early in the activity, students may have problems determining the area correctly. A brief discussion about dissecting the figure into simple rectangles and triangles will help them through this phase of the investigation.

For example, you might have students determine the area of the triangle below by considering the area of square surrounding the triangle and subtracting the areas of the unnecessary pieces:

2083 inscribed tri decompose 

The area of the square is 4 square units, and the total area of the surrounding pieces is 2.5 square units. Therefore, the area of the triangle is 4 – 2.5 = 1.5 square units.

Distribute the Discovering Pick’s Theorem Activity Sheet to help students begin their investigation. Questions 1‑5 serve as the basis for the investigation, whereas Questions 6‑14 help the investigation along step-by-step. (You may wish to distribute only the first page of the activity sheet to students at the beginning of the lesson; the second and third pages can be distributed later, as students are ready for them.)

pdficon Discovering Pick's Theorem Activity Sheet 

The worksheet is designed for students to move back and forth between personal conjectures and formalizing statements within a group. Having groups also allows students to double-check their area calculations with one another.

By the end of the activity, students should have the formal equation for Pick’s Theorem, A = ½P + I – 1.

References

  • Russell, R. A. (2005). "Pick's Theorem: What a Lemon!" Mathematics Teacher 97 (5), 352‑356.
  • Coxeter, H. S. M. (1961). Introduction to Geometry. New York: Wiley.

Assessment Options 

  1. Use the Discovering Pick's Theorem Activity Sheet as a form of assessment.
  2. Students may write reflections about the discovery process, their ability to create figures given a certain number of perimeter and interior pins, and how quickly the pattern emerged in their group discussions.

Extensions 

  1. Students tend to appreciate the opportunity to invent multiple figures with a given value for P and I. For instance, students will try to express their creativity by finding a unique figure that has nine perimeter pins and three interior pins. (Some possibilities are shown below.)

    2083 pick P( 

  2. To create an idea of graphing the equation, have students use stacking cubes to create a value for the area. (Because the area occur in intervals of ½, students might want to use two stacking cubes to represent an area of one square unit.) Have the students place these stacks on a set of coordinate axes where P is on the x‑axis and I is on the y‑axis. So, a stack representing 6.5 would be placed on the point (9, 3). This graph will allow students to see how the values increase along a fixed I‑value as well as a fixed P‑value.
  3. On a photocopied grid of dots, have students estimate the area of irregular figures. The figures could range from student drawings of random shapes to outlines of states.
  4. Move on to the next lesson, Pick’s Theorem as a System of Equations.
 

Questions for Students 

1. How can you measure the area of figures with several "unfriendly" sides, that is, with sides that are not horizontal or vertical?

[One option is to use Pick’s Theorem, assuming the figure can be placed on the coordinate plane so that its vertices align with lattice points of the grid.]

2. What happens to the area if we look at several figures with different numbers of perimeter pins, but the same number of interior pins? What is the effect of changing the number of perimeter pins?

[When the number of perimeter pins increases by 1, the area increases by ½.] 

3. What happens to the area if we look at several figures with different numbers of interior pins, but the same number of perimeter pins? What is the effect of changing the number of interior pins?

[When the number of interior pins increases by 1, the area increases by 1.]

Teacher Reflection 

  • Did you use the worksheet? If so, did it provide too much structure? Would you consider using only the first-half of the worksheet in future?
  • Did students need more instruction on finding odd shaped areas? Would you include a refresher of finding the area of triangles in future?
  • Did the students remain actively engaged with the mathematics? How did the use of rubber bands or the computer affect classroom management?
  • Did it become necessary during the lesson to make adjustments to keep the students moving toward the objective? If so, what could you introduce the activity differently to ensure that all students understand the goal?
 
Unit Icon
Measurement

Investigating Pick's Theorem

9-12
Rediscover Pick's 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.
Algebra

Pick’s Theorem as a System of Equations

9-12
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.
Algebra

Rates of Change in Pick’s Theorem

9-12
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.

Learning Objectives

Students will:

  • Investigate several geoboard figures to determine the pattern between the number of perimeter pins, the number of interior pins, and the resulting area.
  • Create an equation using symbolic algebra to represent the pattern.

NCTM Standards and Expectations

  • Generalize patterns using explicitly defined and recursively defined functions.
  • Interpret representations of functions of two variables.
  • Make decisions about units and scales that are appropriate for problem situations involving measurement.