The sequence of the Unit and Implementation
Keeping in mind that my 3rd graders need to have experiences in which they can manipulate the spaces they measure to construct deep understanding, the activities and teaching strategies proposed in this unit will include the use of manipulatives and “real life” experiences.
The focus of the unit is to address the misunderstanding that rectangular figures with the same perimeter must have the same area and vice versa. It is not rare that students in elementary grades think that when the area of a figure decreases or increases, the perimeter will also decrease or increase. Students don’t realize that it is possible to have many rectangles with the same area, but different perimeters. On the contrary, if the perimeter is the same in a set of rectangles, the area of those rectangles does not have to be the same. For example, rectangles with side lenghts 2 by 4 and 1 by 8 both have an area of 8 square units, but their perimeters are 12 units and 18 units respectively. Likewise, rectangles with the same perimeter can have different areas. For example, 3 × 4 and 2 × 5 rectangles with side lengths 3 by 4 and 2 by 5 both have a perimeter of 14 units, but their areas are 12 square units and 10 square units respectively.
Through the activities in this unit, students will be able to generalize about the relationship between area and perimeter. For example, given a rectangle with whole-number side lengths, the dimensions are factors of the area. When the difference between the dimensions of a rectangle with a given area is the smallest, you will have the smallest perimeter. When the difference between the dimensions of a rectangle with a given area is the largest, you will have the largest perimeter. Given a fixed perimeter, the rectangle with the largest area will be the one with the dimensions that are closest together (a square). Given a fixed perimeter, with whole-number side lengths, the rectangle with the smallest area will be the one with the dimensions farthest apart.14 One of the side lengths will be 1. During the activities of this unit, although students will tile rectangles of various dimensions to determine area and perimeter, and will make generalizations about the relationships between area and perimeter, the focus is to experiment with non-rectangular figures to better understand the differences between the two concepts.
Beginning with a particular rectangle, students will explore different possible shapes obtained by eliminating one unit square at the time. This type of exploration will generate a variety of options for students to analyze and generalize. In all cases, students will keep their new arrangements within the area of the original rectangle, also known as “enclosing rectangle” by making sure students do not eliminate an entire row or column of the original rectangular array. The sequence of activities will culminate with a “real world” experience in which students will need to use the school hallways as “enclosing rectangles” to arrange possible options to design sensory paths.
The activities included in this unit will require the use of tiles as manipulatives as well as drawing on graph paper and recording results in charts.
The figure below represents some examples of possible arrangements that students may find when manipulating rectangles by eliminating subsequent unit squares. Not all the possible arrangements have been included in the figure.
Figure #3: Example of unit square elimination to differentiate area and perimeter. Not all possible arrangements are represented here. As the number of possible arrangements will be quite large, I recommend to restrict this activity to small rectangular arrays.
The sample chart shown below could be used for students to record their findings as they eliminate one unit square at the time within the dimensions of the original rectangle. Each elimination will require a chart.
Figure # 4: Sample chart for recording area and perimeter of resulting arrangements during unit elimination activity.
Comments: