You manage parks for Corvallis, and you are comparing two rectangular parks that have the same total area. Park 1 is 2 km long, while park 2 is 3 km long. Is the width of park 2 greater, less than, or equal to the width of park 1?
- Building Blocks: If \(l\) is the length and \(w\) is the width of a rectangular park, the area is \(A_{rectangle} = lw\text{.}\)
- Claims: Since the area of the two parks is constant, we can say that \(A_1 = A_2\) or alternatively \(l_1 w_1 = l_2 w_2\text{.}\)
- Reasoning: Since \(l_2\) is greater than \(l_1\text{,}\) we must have that \(w_2\) is smaller than \(w_1\) to keep the product of l and w the same for the two parks. The width of park 2 is less than the width of park 1. A way of making sense of this is to draw pictures of two example parks that have different widths but the same area.

Commentary
- Building Blocks: The Building Block for this problem is relatively simple! Sometimes you will only have one simple Building Block like in this problem, while other times you may have several.
- Claims: An important claim here is that the parks have the same area. Note that even though this is stated in the problem, make sure you list it here—often there is information in the problem statement that is not directly relevant or necessary to consider.
- Reasoning: Note that you should go through every step of the reasoning here, starting from our Building Block: first, from the claim that two areas are equal, use proportional reasoning to go from the given relationship between \(l\)’s to the desired relationship between \(w\)’s. You can mix words, symbols, and even pictures in your reasoning!