Finding Dimensions and Maximizing Areas of Rectangles with Fixed Perimeter
Vittoria Macadino
3/13/11
Algebra 1: 8th - 9th Grade
Overview:
The purpose of this activity is for students to use problem-solving techniques to find the unknown dimensions of a rectangular garden when given its area and perimeter, and also to develop a method for maximizing the area of a rectangular dog fence with fixed perimeter. Students will be able to apply their knowledge of writing and solving linear as well as quadratic equations to the context of a real world application problem involving area. My intent in providing students with possible strategies and manipulatives for solving the fencing problem is for students to choose how to effectively integrate these strategies to develop and explain their own method of problem-solving. My aim is for students to be able to present their knowledge in a variety of ways, including graphically, pictorially, algebraically, or using a table.
Content:
- Writing and solving a system of linear equations to model real-world application problems.
- Writing and a quadratic equation to model area
- Finding the vertex of a quadratic function
- Graphing quadratic functions
- Word problems involving dimensions, area, and area perimeter of rectangles
Objectives:
- Students will apply their knowledge of writing and solving linear and quadratic equations to the context of a real world application problems involving area.
- Students will analyze and solve problems in which rectangles with identical perimeters are compared to maximize area.
- Students will analyze problems by collecting data and searching for patterns and generalizable relationships.
- Students will represent problem situations with models.
- Students will find fixed perimeter rectangles with maximized area using qualitative and quantitative analysis.
- String to represent a fixed length of fencing
- Graphing calculators
- Graph paper
- Handout with problems
Problems/Activity:
Fencing Problems: Finding Unknown Dimensions and Maximizing Area with Fixed Perimeter
You may work in groups of two or three to solve the following problems.
1) You work for a fencing company. A customer called this morning, wanting to fence in his 1,320 square-foot garden. He ordered 148 feet of fencing, but you forgot to ask him for the width and length of the garden. Because he wants a nicer grade of fence along the narrow street-facing side of his plot, these dimensions will determine some of the details of the order, so you do need the information. But you don't want the customer to think that you're an idiot, so you need to figure out the length and width from the information the customer has already given you. What are the dimensions?
2) Another customer calls with a different request. He wants to create a fence for his dog, who has run away a couple of times and often runs into the neighbor’s property. He has ordered 148 feet of fencing, but he wants to know what dimensions the rectangular pen should be to allow his dog the maximum amount of space to run around and play. Each side of the rectangular pen must be a whole number in length. Design a rectangular pen for the customer’s dog meeting his requirements. You may use any of the following methods (a combination of these methods, or your own method) to solve the problem, but you must EXPLAIN your procedure for arriving at your solution and you must include at least one diagram.
a. Draw a labeled diagram to model the problem. You may use string to assist you in modeling the problem. (Each group is allowed 148cm).
Experiment with at least 5 different rectangles.
b. Make a table to record the dimensions (length, width, perimeter, and area) for each fence model.
c. Develop an algebraic equation to model the area of the rectangular pen. Make a graph of your function.
d. Explain your procedure for arriving at your solution.
Assessment:
A rubric will be used to evaluate understanding of the the topics involved (i.e. how to find unknown dimensions of a rectangle and how to maximize the area of a rectangle with fixed perimeter) and the logical structure of the explanation.
Modifications (Appropriateness for Intended Grade Levels):
This activity can be adapted for different grade levels and ability levels. Students in Grade 8 Algebra 1 or Pre-Algebra who do not yet have an understanding of quadratic equations can use the first two methods (a and b) to maximize the area; whereas, students in Grade 9 Algebra I who do have the necessary background knowledge of linear and quadratic equations, can use the third method. It is a challenging problem-solving activity for both age groups, because the suggested strategies provide some direction, but there is no set procedure for solving the problem. It is up to students to integrate and experiment with these methods to develop and explain their own problem-solving technique. The opportunity to present knowledge and information using a variety of methods (graphically, pictorially, in writing, and through tables) meets the needs of a diverse group of learners with multiple abilities.
Adapted from the following sources:
Stapel, Elizabeth. "Geometry Word Problems: Complex Examples." Purplemath. Retrieved 13 March 2011 from
http://www.purplemath.com/modules/perimetr4.htm.
Armin, Tim (January, 2011). Maximizing and Minimizing the Area of Rectangles Given a Fixed Perimeter. In Smile Program Mathematics Index, Illinois
Institute of Technology. Retrieved 13 March 2011 from http://mypages.iit.edu/~smile/ma9601.html.