Perimeter, Area, Volume, and All That: A Study of Measurement

General Learning Strategies

The strategies which will help students achieve the overall learning goal include the use of manipulatives, visualization through project-based learning, and direct-instruction paired with both open-ended tasks and open-ended prompting. In this curriculum unit, open-ended activities and project-based learning should draw out prior knowledge of the properties and definitions of rectangles and expand student understandings of these properties before they begin processes of measurement in later activities. However, having open-ended tasks and using their own prior knowledge, some students may choose to measure their shapes early on with rulers. Research has shown that strategies such as increasing spatial ability through manipulatives and through student-generated illustrations can also increase student performance and understanding.⁵ Therefore, students will use a variety of manipulatives as unit squares to create and to draw rectangles with various perimeters and areas. They will do so in order to strengthen foundational understandings of perimeter and area measurement, specifically regarding the comparison of rectangles with fixed perimeter or fixed area. In creating visual representations of their rectangles, students will be able both to compare the measurable characteristics of their rectangular shapes and form a spatial understanding of rectangular objects. Students will, for instance, have ample opportunity to draw various rectangles from written prompts and then to practice measurement of perimeter and area. The method of creating rectangular shapes out of whole unit grids will then be connected to previously learned knowledge of arrays and multiplication. This will be imperative in developing a sufficiently strong conceptual understanding of two-dimensional measurement so that students can adeptly manipulate formulas. It will also lead students to the idea that deriving area of rectangles is comparable to deriving area of other two-dimensional shapes.

The development of taxonomical frameworks of two- and three-dimensional shapes will also be helpful in having students compute measures in this unit. In the sixth-grade Virginia Standards of Learning, there is relatively little emphasis on the taxonomy of common shapes into typologies.⁶ Nonetheless, students should come to understand that rectangles are a subset of polygons which have reflectional symmetry of two degrees and with diagonals that have equal lengths.⁷ While not often emphasized, such ordering is quite instrumental in developing connectional knowledge of shapes. In Teaching Children Mathematics, Siegel and Ortiz describe the need for correct sorting of shapes before moving into measurement application.⁸ They explain that before discussing perimeter with elementary students, they reviewed classification because “some students had struggled naming quadrilaterals, I first asked the class to name the polygon.”⁹ In the case of Siegel and Ortiz, the relatively informal process of identification was done with a “thumbs-up” math conference and produced the satisfactory result of students coming to the idea of multiplying both measures by two and then adding the products together to find a rectangle’s perimeter.¹⁰ While Siegel and Ortiz had moderate success in such a review, this success may not be met in classes with students with disabilities who may have impairments in cognitive processing or memory retrieval. As this is the case, this curriculum unit will include direct-instruction and scaffolded elements that reteach classification as opposed to informal review. Additionally, it will use graphic organizers such as those featured in Teaching Student-Centered Mathematics to explore patterns in area and perimeter.¹¹ More specifically, during practice in categorization, the square will be presented as having a reflectional symmetry across four different lines, while the rectangle will be presented as having two lines of symmetry.

Figure 1. This illustration depicts a square’s four lines of symmetry and a rectangle’s two lines of symmetry.

Another significant process in creating foundational understandings of area measurement is having students create grids on squares of the same length and width and then having them count the units. This procedure fits within the larger strategy of having students explore concepts with multiple means of representation and can also be connected to previously learned knowledge of addition, arrays, and multiplication. Alternatively, it can be done through the open-ended task of having students create polyominoes out of two-dimensional unit squares. Students will then focus on creating polyominoes that are nets of three-dimensional figures. According to Van de Walle and Lovin, it is important to use whole number units and to provide students with rulers and unit squares while doing this.¹² As polyominoes are iterations of a single unit square (i.e. each tile used to make a polyomino has the same dimension), they are versatile tools in teaching operations such as addition and multiplication and, also, in teaching geometrical concepts such as distance, perimeter, and volume. Additionally, Van de Walle and Lovin explain that the process of students’ using rulers and a square unit to determine area without explicit instruction is critical in developing a strong conceptual understanding of two-dimensional measurement.¹³ The following quote explains that instructors should guide students’ own determinations toward the main idea of area as a measurement of arrays before introducing formulas:

“As students discuss their various approaches, focus on those ideas that are closest to this idea: The length of a side will determine the number of squares that can be fit on the side. The length of the other side will determine how many rows of these squares will fit in the rectangle all together. Multiply the length of a row by the number of rows.”¹⁴