TIME ALLOTMENT:
One to two class periods

OVERVIEW:
How is the area of rectangles affected if the perimeter stays constant? Students will explore the effects on the areas of rectangles with constant perimeters. They will model rectangles on grid paper and represent area as a product of factors. Students will also verify findings by looking at the relationship between length and area graphically.

SUBJECT MATTER:
Plane Geometry, English Language Arts

LEARNING OBJECTIVES:
The student will:
• Find the areas of rectangles with constant perimeters.
• Apply concepts of geometry to real-world problems.
• Find maximum area possible for a given perimeter.

MEDIA COMPONENTS:
Geometry Journey (World of Geometry) Video 1: Plane Geometry
START at 3:45

Geoboards in the Classroom
http://mathforum.org/trscavo/geoboards/geobd4.html
This website allows students to experiment with geometric pattern making.

MATERIALS:
Per each student:
• grid paper
• graphing calculator
• story problems HTML   PDF

STANDARDS:
National: Principles and Standards for School Mathematics
http://Standards.nctm.org/document
Analyze characteristics and properties of two- and three-dimensionalgeometric shapes and develop mathematical arguments about geometric relationships.

State: Louisiana Mathematics Framework Bulletin
http://www.lcet.doe.state.la.us/doe/assessment/standards/MATH.pdf
In problem-solving investigations, students demonstrate an understanding of geometric concepts and applications involving one-, two-, and three-dimensional geometry, and justify their findings.

G-1-H: Identifying, describing, comparing, constructing, and classifying geometric figures in two and three dimensions using technology where appropriate to explore and make conjectures about geometric concepts and figures;
G-2-H: Representing and solving problems using geometric models and the properties of those models (e.g., Pythagorean Theorem or formulas involving radius, diameter, and circumference).

PREP FOR TEACHER:
• Prior to lesson, students should have a basic understanding of perimeter and area. Students should be able to recall the formulas for both and understand the dimensions of a rectangle. Show students the video “Geometry Journey”. (CUE and START at the beginning of the segment “Areas of planar figures” STOP the video at the end of the segment.
• Students should also be familiar with graphing calculator and keystrokes.

Introductory Activity:
1. Arrange students in groups of 3-4. Ask students to designate one person as recorder.
2. Give each group copies of the story problems as well as grid paper.
3. Provide a graphing calculator for each group.

LEARNING ACTIVITIES:
1. Have students read first story problem. Reinforce vocabulary terms introduced earlier.
2. Discuss the formulas for perimeter (P = 2L + 2W) and area (A = LW). Model any rectangle with a perimeter of 24 square units on grid paper. Ask students to investigate the relationship between the length and width. (Note: In this instance, the sum of the length and width is 12.)
3. Have students find the area of the rectangle modeled. Ask students in each group to model as many rectangles with integral dimensions as possible with a perimeter of 24 units. Have students find the area of each rectangle and record the lengths, widths, and areas on the data sheet.
4. In each small group, have students discuss the results of their findings. For which rectangle is the maximum area found? What is this rectangle more commonly called?
5. Read the second story problem. Ask students to repeat steps 2-4 with the new perimeter. Which rectangle yields the maximum area? Is this consistent with what happened the first time?
6. Ask students to find the perimeter of a rectangle that yielded a maximum area of 64 square units. Have students discuss in small groups the strategies they used to determine the perimeter. Ask students if the work they did previously helped them to make this determination. (Note: Students should recognize that a square yields the maximum area.)
7. Repeat step 6 using areas of 81 square units, 100 square units, and 121 square units. Ask students to choose an area on their own and use whatever strategy they previously used to find the perimeter.
8. Ask students to use the lengths and areas found from one of the story problems and record the data in lists in the graphing calculator. The lengths should be in List 1 and the areas in List 2.
9. Make a scatter plot of the data. Let List 1 represent x and List 2 represent y.
10. Have students in groups determine if the data is generally linear or quadratic. Ask students to do the appropriate regression and find the r-value. (Note: Make sure the diagnostic function is on.)
11. Ask students to analyze the graph to find the maximum value for area. What value of the length yields the maximum area? What type of rectangle is this? Is this consistent with previous findings?

CULMINATING ACTIVITY:
1. Write a summary of the lesson regarding the maximum area yielded by rectangles of constant perimeter.
2. What happens if the gardener uses the side of his house as one side of his garden? Use a perimeter of 24 feet. Will the same results occur?

CROSS-CURRICULAR EXTENSIONS:
Social Studies/Economics: Relate how changing the area of house to be constructed will have an effect on the cost of building and selling.
Art/Physical Education: Design a sports field with the same area as a football field or basketball court but with a different shape.
Art: Give the students an area for the desired living space for a one story home that is to built and have them design the floor plan.

COMMUNITY CONNECTION:
Compare the blueprints of different school system facilities such as cafeterias or gymnasiums.

STUDENT MATERIALS:
Student handout: Data Sheet HTML   PDF
Student handout: Story Problems 1 and 2 HTML   PDF   (Answers HTML   PDF )