I designed this activity as an opportunity for a calculus student to build an intuitive understanding of why the product rule works.

While completing this activity, students develop their intuition by exploring a familiar topic (the area of a rectangle).

During this activity, students are led to make the following observations:

1. When the length of a rectangle increases, and the width stays the same, the increase in area equals the increase in length times the width.

2. When the width of a rectangle increases, and the length stays the same, the increase in area equals the increase in width times the length.

3. When the length and width of a rectangle both increase, the increase in area equals the increase in length times the current width plus the increase in width times the current length.

By the end of this activity, students are expected to have discovered a formula for determining the change in rectangle area:

Change in Area = (Change in Length x Width) + (Length x Change in Width)

A teacher or professor could use this activity as a pre-lecture exercise (to build intuition ahead of time) or as a post-lecture exercise (to reinforce the product rule concept and allow students to connect it to a familiar situation).

Also, this activity could be used as a tool to teach the product rule for the first time or as a tool to teach the product rule in the context of related rates.