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Relationships Between Area and Perimeter

 Activity Description Activity Guide

• Discuss the following scenario in small groups.

Scenario: (from Ma, L. 1999. Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum, p. 84)

Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing:

• How would you respond to this student?

• Is this student’s “theory” valid? After initial discussions in small groups, write a short journal entry about your reaction to this student and a brief mathematical discussion either validating or disconfirming this “theory.”

Part 2: Exploring perimeter with a fixed area with manipulatives

• With 12 square tiles, arrange the tiles (flat on a surface) so as to obtain the maximum perimeter. The tiles should touch completely side-to-side, not by corners.

Examples include:

• What is the maximum perimeter? What general observations did you make about how to make your figure have a larger perimeter? Is there more than one figure that can have the same maximum perimeter? Why or why not?

• Restrict the figure to a solid rectangle. Find all possible rectangles with an area of 12 square tiles.  Record the respective dimensions and perimeter for the rectangles. Which rectangle has the maximum perimeter? Why?

• Given any fixed area for a rectangle, make a general statement about which dimensions maximize the perimeter.

Part 3: Exploring area with a fixed perimeter using manipulatives

• Consider the following problem:

Farmer Ted wants to build a rectangular pigpen for his pigs.  He has 24 meters of fencing left from his last project. What size rectangle should he build so that the pigs have the maximum amount of play area?

• Use square tiles to model different-sized solid rectangular pigpens. Build all possible rectangular pigpens that can be enclosed with 24 meters of fencing. Record the corresponding dimensions and areas for those pigpens. Which pigpen gives the pig the most play area?

• As the length of the rectangular pigpen changes, how does that affect the width and area of the pigpen? What patterns do you notice? Explore this relationship numerically and express this algebraically. Use this relationship to explain why the maximum area occurs with a square pigpen.

• Given any amount of fencing, make a general statement about the dimensions of the rectangular pigpen that give the maximum area.

• Open the Excel file called maxarea.xls. (You will need to ENABLE MACROS when prompted.) This spreadsheet allows the user to change the perimeter (P) and then vary the length of side of X using the scroll bars.  As these values change, the yellow box will display the current values of P, X, Y, and the Area (A) of the rectangle. The rectangle and the data point (X, A) on the graph of the Length of side X vs. Area will also change accordingly.

• Fix the perimeter (P) to 24 as in the pigpen problem. Then incrementally increase the length of side X from 1 to 12. Describe the changes that occur in the numerical, graphical, and geometrical representations.

• Click on the Show All Points tab to display the next worksheet. What is the shape of the graph of all the data points (X, A)? Why does this make sense? Which Length of X gives the maximum value for A? Why?

• How does this experience differ from the manipulative experience of physically building each rectangle? Discuss the benefits and drawbacks of the physical and spreadsheet approach to the pigpen problem.

• Do the same exploration with the following perimeters and record the results in a table. Make a prediction of the dimensions and maximum area before using the spreadsheet to simulate the changes in the rectangle for each perimeter.

• If Farmer Ted had 158 meters of fencing, predict what the dimensions would be of the rectangle with the maximum area.  Explain how you made this prediction. Create a rule as a function of P to obtain the length of side X which gives the maximum area. Confirm this rule with several other examples.

• Create a general equation for determining the Area of a rectangle given both a fixed P and length of side X. How does this equation relate to the shape of the graph in the spreadsheet? With this Area equation, use the first derivative to verify the value of X which gives the maximum value of A. How does this compare to the rule you created above (a function of P to obtain the length of side of X which gives the maximum area)?

• Discuss the non-calculus and calculus based approaches to making this generalization, including students’ accessibility, advantages, and disadvantages.

• Why is the maximum value for Area achieved when the first derivative is equal to zero? For a graphical exploration of this, use the following java applet:

http://www.ies.co.jp/math/java/calc/doukan/doukan.html

Part 6: Revisiting the Opening Scenario

• Revisit the scenario and subsequent group discussions from Part 1. Now that you have explored the relationship between perimeter and area at several different levels and with various tools, how would you use the student’s “theory” to provide relevant classroom experiences if you were teaching:

1.      A sixth grade class

2.      An Algebra I course

3.      A Calculus course

Either discuss this in small groups or write an individual journal entry that addresses how you would approach this topic at each of the three levels, including justifications as to why you believe your approach is pedagogically and mathematically appropriate.

Extensions:

• Consider the generalization you made in Part 2 when a rectangle has a fixed area and you want to maximize perimeter. Create an interactive spreadsheet template that shows the numerical and graphical representations of this problem. Use scroll bars to fix the value of A and change the values of X, and create a graph of X vs. P.

• View the following java applet:
http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/calculus.dir/max-min.dir/ problem1.html
Discuss the benefits and drawbacks of using the applet or the spreadsheet template to explore the relationship between perimeter, area, and the length of side X.

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