The goal of this lesson is for students to use the spreadsheet as a tool for observing patterns over a large range of cells. Since the area depends on two variables—perimeter pins and interior pins—the spreadsheet representation is a natural.
When using a physical geoboard, students will naturally focus on perimeter pins over interior pins because increasing the number of perimeter pins is fundamentally easier than increasing the number of interior pins. When using a spreadsheet, students traditionally see the effect of interior pins first.
Begin by having a group of students create a spreadsheet with the number of perimeter pins along the top row and the number of interior pins along the first column. A nice discussion arises when students realize there are no figures with 0, 1, or 2 perimeter pins. So, they may begin perimeter values at 3. Since there are many figures with no interior pins, the column may begin with a value of 0. Students can create a file on the computer using Excel, Quattro, or some other spreadsheet application; or, they can use the Pick's Spreadsheet activity sheet. (The benefit of using a computer spreadsheet program is that once students recognize a pattern, they can fill in the remaining columns automatically.)
Groups may take a random approach to filling out the table of values, or they may be much more diligent in filling out rows or columns systematically. Of course, the more systematic approach leads to an earlier conjecture for the constant rates of change. The spreadsheet below shows the chart when results are organized systematically.
Once students have made the conjecture that row values increase by ½ and column values increase by 1, they may fill in a value on the table and then check to see if a corresponding figure has the predicted area. For example, a student may conjecture from the pattern that a figure with eleven perimeter pins and nine interior pins has an area of 13½ square units. The group members should then find such a figure and prove (or disprove) the conjecture.
Once students have successfully shown the pattern, have the class discuss the rates of change. Each group should focus on formalizing the pattern using symbols. Ask the groups to use A, P, and I to write an equation for Pick’s Theorem. The formalization may be difficult if the students have not completed one of the previous lessons, Rediscovering the Pattern of Pick’s Theorem or Pick’s Theorem as a System of Equations. If the students have completed one or both of those lessons, then the formula falls out quickly. Either way, a good discussion of the transition between row behavior on the spreadsheet and the coefficient of P should be explored.
Because the coefficient represents the rate of change, the coefficient could be represented as r/s where r is the change in area and s is the change in the number of perimeter pins.
Thus, the coefficient of P represents a ½ square unit change in area each time the number of perimeter pins increases by 1. That is,
Similarly, the area changes by 1 square unit each time the number of interior pins increase by 1. So, the coefficient for I is 1. Hence, this gives the formula as A = ½P + 1I. However, students realize that the formula is off by 1 square unit in every case, and the correction requires a constant term of ‑1, giving A = ½P + I ‑ 1.
For an algebra class or precalculus class, the creation of a formula and the discussion of slope is a good stopping point. For a calculus class that is ready for the concept of partial derivative, pose the following question: "What does it mean to have one variable fixed while the other is allowed to change?"
Performing the partial differentiation is part of an introduction to the topic. Given the equation A = ½P + I ‑ 1, if I is held constant, then the partial derivative of A with respect to P is ½. This is the same phenomenon that the students recognized in the spreadsheet. Similarly, by holding P constant, the partial derivative of A with respect to I is 1. As with any analogy, it can be pushed far enough to break down—Pick’s Theorem is not continuous, so partial differentiation is not allowed. But, the hands-on example and basic discussion of holding one variable fixed while another is allowed to change can be effective. And the breakdown of the analogy may generate some nice discussions about continuity.
