Use the following Interactive Activity to explore how the perimeter of rectangles with fixed area varies with the shape of the rectangle. Pay attention to the relationships and connections that you notice.
After you have explored the perimeter of several rectangles with areas of 16 square units, please answer the following questions:
Question: How does this activity encourage the building of connections while solving problems?
The actual length and width of each rectangle are created with square units and then recorded in a table, thus making connections between representations. Also, exploring the relationship between the area and perimeter leads to eliminating inappropriate connections, namely, assuming that the two behave similarly –– the classic misconception is that two figures with the same perimeter should have the same area.
Question: What connections did you make while reasoning about the problem? How did your experience with one rectangle connect to your plan for the next rectangle?
Answers will vary. After drawing two rectangles, a conjecture might be made that long, thin rectangles have greater perimeters. This idea can be tested by actually drawing more rectangles and also by using reasoning.
Question: How does this problem connect to understanding of decimal numbers?
Patterns in the table show that longer rectangles have greater perimeters. When trying to draw the rectangle with an area of 16 and the largest possible perimeter, it is sensible to think beyond whole-number lengths and widths, since the grid allows for rectangles longer than 16 units of length. This may not be immediately apparent. Extending beyond the grid provided in this activity would allow for smaller decimals and thus result in even larger perimeters.