The Study of Area
Age: 8 -12
A child in a Montessori classroom encountering the work of area approaches the concept with an amount of experience that, in some cases, is measured in years. Having worked with the Constructive Triangles sensorially in an Early Childhood classroom, with added nomenclature at Lower Elementary, they have built dozens and dozens of figures with triangles and identified their relationships to each other. Further work, with the Insets of Equivalence, added to their understanding that the area of a figure is directly related to the lengths of its parts, bases, heights, and diagonals.
A good example of the connections between Equivalence and Area can be seen in the first Equivalency Inset, which compares two figures, a triangle, and a rectangle. The triangle, divided into a trapezoid and two small right-angled scalene triangles, can be rearranged to fit the negative space of a rectangle on the right. The child discovers, with their hands, that a triangle will be equivalent to a rectangle if the triangle’s height is twice its height and its bases are equal. Compare this to our eventual area formula for triangles, base times half its height!
Still, more connections can be seen. The Study of Area is logically followed by the Study of Volume. Each will utilize a “mediator” figure, one that a child can easily calculate. In the case of Area, we’ll use the rectangle (just count the squares, base times height). The child carries that experience to the more abstract Study of Volume when the mediator becomes a rectangular prism (just count the cubes, length times width times height). And finally, a nice integration with Arithmetic is available to us here. What is Squaring if not Area? What is Cubing if not Volume? In this way, Geometry informs Arithmetic and Arithmetic informs Geometry.
