In this activity students calculate the area of an irregular shape presented in a scale drawing. In this case, students can calculate the area exactly by composing and decomposing triangles and parallelograms. As students choose an appropriate way to compose and decompose the floor plan of the house in order to make efficient calculations.

Monitor for students who use these different strategies to determine the area of the floor plan:

• Decompose the floor plan into lots of small rectangles and a triangle and then add their areas, for example 

  

• Decompose the floor plan into a few, larger rectangles and a triangle and then add their areas, for example

  

• Do a combination of some decomposing and adding with some surrounding and subtracting, for example 

  

• Enclose the floor plan with a large rectangle, subtract the missing pieces, and then add on the balcony

  

Plan to have students present in this order to support moving them from less efficient to more efficient solution methods.

The purpose of this discussion is to highlight different ways of decomposing the figure and show that they result in the same answer for the area of the region.

Invite previously selected students to share how they calculated the area. Sequence the discussion of the strategies in the order listed in the activity narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How are the strategies the same? How are they different?”
• “Why do the different strategies lead to the same outcome?”
• “Are there any benefits or drawbacks to one strategy compared to another?”

The key takeaway is that any combination of decomposing, adding, enclosing, and subtracting will work to find the area of a complex region, as long as the area of each subpart of the region is accounted for once and only once. The method will be more efficient if students can find ways to combine multiple, smaller sub-regions into fewer, larger sub-regions.
 

In this activity, students use strategies developed in earlier work to estimate the area of the state of Nevada. In this case, the shape is not a polygon, so the area can only be estimated.

First, students identify the information needed to estimate the area of the state from the given map. Next, they choose how to model the region with polygons in order to estimate its area (MP4). Monitor for students who:

• Decompose the image of the state into a rectangle and a right triangle, find the area of each, and add the two.
• Enclose the image of the state in a rectangle, find the area of the rectangle, and subtract the area of a right triangle.

These two methods work equally well for this shape. Also, some students may notice and account for the missing area in the southeast corner of the state and others may not.

If students decompose the image of the state into a rectangle and triangle, they may use the 270 miles for a side length of the rectangle instead of finding the difference of the 490 miles on the opposite side and 270 miles. Ask them to check their answer with a partner and to reevaluate their calculations.

The goal of this discussion is for students to understand the distinction between calculating the areas of geometric objects and estimating areas of regions on maps.

First, display these questions for students to discuss with their partners:

• "How did you make your estimate?"
• "If your estimates are not the same, are they close? What accounted for the difference?"

Ask students how finding the area of Nevada in this activity was the same as finding the area of the floor plan in the previous activity and how it was different.

• One way it was the same was that it was still helpful to decompose the region into rectangles and triangles. Strategies involving addition and strategies involving subtraction were both possible.
• An important difference is that the state is not a polygon. Some of the boundaries are not straight and the overall land is not completely flat. Assuming the state is flat and approximating the boundaries with line segments both lead to some error in the estimate.

Consider telling students that the actual area of Nevada is about 110,560 square miles.