Optimizing a Rectangle

Overview: Students determine the optimal swimming area for a fixed swimming perimeter, selecting from a variety of tools/resources to support their investigation.

Curriculum Expectation(s):

• Linear Relations: apply data-management techniques to investigate relationships between two variables
• Measurement and Geometry: determine, through investigation, the optimal values of various measurements of rectangles

Learning Goal(s):

• To investigate the area of different rectangles given a fixed perimeter, where one side is bounded by an obstruction
• To determine the maximum area of a rectangle with a fixed perimeter, where one side is bounded by an obstruction
• To use data management strategies to organize data

Timing and sequencing: This 75-minute activity was created as the third in a series of activities on determining the optimal values of various measurements of rectangles. One suggested order is as follows, with the teacher scaffolding the first activity and then encouraging greater independence as students proceed through each activity:

1. Determine the maximum area of a rectangle with a fixed perimeter
2. Determine the minimum perimeter of a rectangle with a fixed area

Classroom constructs: Students work in small groups (groups of two or three)

Materials:

Group work

• String
• Tiles
• Grid/Graph paper

Individual work

• Blank paper
• Lined paper
• Grid/Graph paper
• Graph paper with 1st quadrant

Description:(click on this link to see the website of this activity and to access all resources)

Teacher will post the following problem:

You have 48 m of rope to make a rectangular swimming area. The shore will be one side of the swimming area; so only three sides of the rectangle will be roped off.

In groups: VNPS or chart paper

Available materials for use: string; tiles; grid/graph paper

In groups, students will explore the problem, including:

• Finding some swimming area (rectangles) that will work.
• Making a conjecture (Predict)
• Making a plan to confirm/disprove their conjecture.

There is a prepared page for students to work on. Alternatively, the prompts may the displayed on a projector/written on the board and students can brainstorm with a VNPS or chart paper.

Individual: Students may use notes from previous investigations

Available materials for use: blank paper; lined paper; grid/graph paper; graph paper with 1st quadrant (provided)

Individually, students will execute the plan to confirm/disprove their conjecture.

There is a prepared page for students to share their work and thinking.

Possible Alternatives/Extensions:

This format can be applied to the case where the perimeter involves two sides only

(see resource provided).