```Maximizing and Minimizing the Area of Rectangles Given a Fixed PerimeterTim Amrein                     Franklin Fine Arts Center                               225 Evergreen St.                               Chicago IL 60660                               (312) 534-8510Objectives:Grade Levels 5-8.  (This activity can be simplified for younger, less mathematically mature students.  Numerous extensions can be added for more advanced students.) (1) Students will analyze and solve problems in which rectangles with identical     perimeters are compared to maximize or minimize area. (2) Students will analyze problems by collecting data and searching for patterns     and generalizable relationships. (3) Students will represent problem situations with models.(4) Students will analyze fixed perimeter problems using x,y coordinate     graphing. (5) Students will find fixed perimeter rectangles with maximized or minimized     area using qualitative and quantitative analysis. Materials Needed:(1) A fixed length of ribbon, or string, which will be used to represent a fixed    length of fencing(2) Paper or cardboard rectangles of given, fixed perimeter(3) Tiles, ceramic or paper(4) Inch tiles (tiles with 1 sq. in. area)(5) Scientific calculators(6) Graph paper(7) Handouts with problems involving fixed perimeter and, if students have prior     experience with it, fixed area. Optional materials:(1) Fixed lengths of actual fencingStrategy:(1) Students will be given 3 paper rectangles with identical perimeters (such as     5 in. by 25 in., 10 in. by 20 in., and 15 in. by 15 in.) They will     additionally be given 5 in. tiles (square tiles whose sides are each 25 sq.     in.)  Give the following instructions and questions:  (1) Use your ruler to     find the perimeter of each rectangle.  (b) What do these rectangles have in     common?  (c) Which rectangle requires the most tiles to completely cover it?     (d) Which rectangle requires the fewest tiles to completely cover it?      (Students will work in pairs)                  (2) Next, the students will be presented with this problem, "You have a plot of     land and a dog.  Your dog has run away a couple of times and often runs on     your neighbors' property.  You decide to fence in a rectangular section of     your land so that your dog doesn't run away but has room to play.  You have     72 feet of fencing.  You want each side of the rectangular "pen" to be a     whole number in length.  Your goal is to allow your dog the maximum amount     of space to run around and play.  Design the rectangle that achieves this     goal."  The students will model this problem using a length of string or     ribbon 72 cm. long.  Students are to experiment with at least 5 different     rectangles.  They are to record the dimensions (bottom edge, side edge,     perimeter, area) for each of their fence models.  We will then discuss the     fact that, geometrically speaking, we are maximizing area given fixed     perimeter. (Students will work in pairs or in groups of four) (3) Next, the students will be presented with these two problems:    (i) "You run a business that puts on banquets. For one small banquet,         you need to seat 12 people.  You construct your banquet tables         from small square tables (which individually seat 1 person on each         side).  Each small table costs your company \$1 per day (for rental         or moving).  Your banquet tables are always rectangular.        (a) What are the dimensions of the table that will seat these            12 people most cheaply?        (b) What are the dimensions of the table that would seat the 12 people            in the most expensive way possible?   (ii) The same basic problem will be repeated for a banquet in which         24 people need to be seated.    For both of these problems, charts will be compiled in which the dimensions    (bottom edge, side edge, perimeter, and area) are recorded for all possible    perimeter of 12 and perimeter of 24 rectangles.    Fixed perimeter coordinate graphs will be completed recording the bottom     edge and area ordered pairs (separate graphs for P = 12 and P = 24).    The shape of these graphs and the information they give will be discussed.    (The points can be connected to form parabolas.  The area optimizing square    and the two area minimizing rectangles will be evident on the parabolic     curve.)   (Students will work in pairs or groups of 4)(4) Similar problems to the first 3 will be given.  The students will be allowed    to use models for some.  For some problems they will not use models.    Use of the perimeter and area formulas will be discussed.  Use of the     calculator (the squaring key, for example) will be discussed.    Students may be asked to devise some of their own problems applying these    concepts to realistic situations.  (Students will work individually.)(5) If the students have experience with fixed area, varying perimeters     problems, this problem type will be further explored.Performance Assessment:Students will be given a problem (such as a fencing or border problem) involving fixed perimeter and maximization and/or minimization of area.  The problem may involve finding an efficient way of doing something.  (Prices of tile per square unit of tile may be given, for example.)  Students will be asked to solve the problem, using models and various forms of analysis.  Graph paper, tiles, string, rulers, etc. will be provided.  Students are to give their solution mathematically, pictorial, and in a paragraph.  A rubric for evaluation will be devised.  Understanding of the topic (the generalizable relationships) and logical structure of the explanation will be the central concern of the rubric. Conclusions:     I have used this activity and variations of this activity with 6th and 7thgraders (and will probably use a version of it with fifth graders next year).It is an excellent problem for developing problem solving ability.  Studentslearn to model, analyze, and represent problems in numerous ways.  The problemalso necessitates the search for patterns and the discovery of rules and relationships, which are vital elements of mathematics at all levels.References:Main source:Fitzgerald, et. al.  Middle Grades Mathematics Project:  Mouse and Elephant:
Measuring Growth.  Addison-Wesley Publishing Company. 1986.Secondary sources:Fitzgerald, et. al.  MIddle Grades Mathematics Project:  Similarity and
Equivalent Fractions.  Addison-Wesley Publishing Company.  1986.Mayfield, Karen, & Whitlow, Bob.  Equals Investigations:  Flea-Sized Surgeons
Regents of the University of California. 1994.

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