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Lesson Plan

Investigation - Rectangular Gardens


Subject

Math, Science & Technology

Grade Levels

Intermediate, 6th Grade


Materials

Grid paper, square tiles, scissors

Authors

The Core Curriculum Companion For The New York State Mathematics Resource Guide Writing Team

Extending The Activity

Use 36 tiles and grid paper to draw rectangles whose area is 36 square units. Explain how to maximize and how to minimize the perimeter.

Assessment 1

What is the largest possible perimeter, using integers, for a rectangular region with an area of 16x2?

a. 34x
b. 20x
c. 16x
d. 8x

Ans. a (The dimensions of the rectangle are x by 16x).

Assessment 2

Joshua said that his rectangular garden has a perimeter of 24 meters.
Kristina said that her rectangular garden has a perimeter of 26 meters.
If both have maximized the area, whose garden is larger? Explain.

Ans.

Kristina has the garden with the largest area. The dimensions of her garden that maximizes the area is 6 x 7. The dimensions of Joshua's garden is 6 x 6.

Additional Notes

For additional information and activities, see pages 109 and 115 in the New York State Core Curriculum/Mathematics Resource Guide.

  • About The Core Curriculum Companion

    • Source

    The Core Curriculum Companion For The New York State Mathematics Resource Guide Writing Team. "Rectangular Gardens". In The Core Curriculum Companion For The New York State Mathematics Resource Guide, 194-197.

    Duration

    1 - 2 lessons

    Procedure

    DOING THE INVESTIGATION

    1. Discuss with your class the concepts of area and perimeter.
    2. Have students use square tiles to create a rectangular garden with a perimeter of 24 feet where the length of the sides are positive integers. Let a side of the tile represent one foot.
    3. Have students record their work on grid paper.
    4. Have students work in groups to create five other rectangular gardens with a perimeter of 24 feet, but with different areas. They should record their work on grid paper and cut the shapes.
    5. Have students arrange the gardens in order from largest area to smallest area.
    6. Pose the following question to your students: Describe the relationship between perimeter and area. Given a perimeter, how can the area be maximized? minimized? Explain.
    7. Allow time for students to try out their conjectures by investigating rectangles with different perimeters, such as a perimeter of 18 units. Explain why your observations are the same or different for a rectangular garden with either perimeter.

    POSSIBLE SOLUTIONS

    In order of size from largest to smallest area, the dimensions (in feet) of the rectangular gardens are: 6x6, 5x7, 4x8, 3x9, 2x10, and 1x11. The areas (in square feet) are: 36, 35, 32, 27, 20 and 11, respectively.
    The closer to a square the rectangle is, the area is maximized (largest).
    The longer the rectangle is, the area is minimized (smallest).
    This observation is verified with a rectangle whose perimeter is 18 feet. Four rectangles can be created whose dimensions are 1x8, 2x7, 3x6 and 4x5. Their areas are 8, 14, 18 and 20 square feet, respectively. Thus, the shape closest to a square (4x5) has the largest, maximum, area.

    Vocabulary

    Rectangle, perimeter, area, maximize, minimize

    Description

    Students will create rectangular gardens with a given perimeter. They will generalize how to maximize the area.

    WHAT'S THE MATHEMATICS?

    • knowing and applying formulas for perimeter and area of polygons
    • understanding length, area, and volume and make relationships between the measurements

    Relates New York City Performance Standards

    • M2a Is familiar with assorted two dimensional objects, including rectangles.
    • M2d Determines and understands length, area, and computes areas of rectangles.
    • M5a Formulation
    • M5b Implementation
    • M5c Conclusion


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