Generalizing Area as "Average" Length x Width

Why are there so many different formulas for area? Many students become dependent on applying formulas, but they forget the formulas soon after the chapter test.

In tutoring, I emphasize that area is ALWAYS related to length by width. I encourage students to think about the middle or average length of a shape. Since the area is a measurement of square units, segments representing length and width must always be perpendicular to each other. For consistency, I have students consider the length as the horizontal measurement, and the width as the vertical measurement (which can also be called the height).

Here are some examples.

Consider parallelograms and trapezoids, as shown by the blue shapes above. If you draw a line segment across a parallogram, parallel to a base, that segment will always be the same length as the base. So, it can be considered the "average" length. For a trapezoid, the middle segment is the same as half of the sum of the two bases. For triangles, as shown in purple above, the middle length is half of the base.

I often have students identify the length and width of a rectangle by counting the squares on graph paper. Then I  extend this activity to identifying and labeling the middle segment of a parallelogram, trapezoid, or triangle. In this way, students are better able to understand the concept of area of these types of shapes and may not need to memorize individual formulas.

In Common Core, students in Grade 2 are expected to partition rectangles into square units (2.G.2). In Grade 3, students count squares to find the area, and relate area to multiplication (3.MD.6 and 3.MD.7). In Grade 4, students calculate areas of rectangles (4.MD.3). In Grade 6, students find area of other polygons (6.G.1 and 6.G.4).

Please write a comment if you have your own special methods for teaching area.