Maximizing and Minimizing the Area of Rectangles Given a Fixed Perimeter

Tim Amrein                     Franklin Fine Arts Center
330 W. 25th Place              225 Evergreen St.
Chicago IL 60616               Chicago IL 60660
                               (312) 534-8510

Objectives:

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 fencing

Strategy:

(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 7th
graders (and will probably use a version of it with fifth graders next year).
It is an excellent problem for developing problem solving ability.  Students
learn to model, analyze, and represent problems in numerous ways.  The problem
also 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.