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

Area and Perimeter

After conceptualizing that rectangles can be decomposed into squares, and after I introduce them to the concepts of area and perimeter, and how to measure them, my students will use what they have learned about the array structure of a rectangle to determine area and perimeter. They can measure both area and perimeter of polyominoes they are presented with, and also of their enclosing rectangles. Students will first need to be introduced to what these concepts are, as well as how they are measured.

“All measurements were converted to linear distances. So it makes sense to start by discussing that most basic, most fundamental of measures: distance.”¹⁵ Perimeter is a type of linear measurement, so I think it makes sense to introduce it first. Students are already learning how to measure length with non-standard units of measure and by using a ruler. When you are measuring distance, you are really measuring how far away the two end points are from each other. First grade students in New Haven get an entire math unit dedicated to working on the skill of measuring the length of different objects; by the time I start this curriculum unit they will have repeated practice in measuring the length of different objects. For this reason, students should be able to easily adapt the skill of measuring length to the skill of measuring perimeter. Instead of just measuring how long something is they will be determining how long the entire outer edge of the object is. Instead of saying a 5×1 pentomino, as shown in Figure 4, is five square tiles long, they should be able to say that this pentomino has two edges length of 5 and two edges length of 1, for a total perimeter of twelve. They will determine this by counting how many edges make up the outside of the figure. Instead of just counting sides of square to find “length” they will be counting to find the length around the object.

Figure 4: Instead of counting how long this design is in terms of square tiles, we are counting the surrounding perimeter by using the edges. There are 12 outside edges in this pentomino therefore it has a perimeter of 12.

I anticipate that area will be a more complex concept for my young students to grasp. “Our brains need help when thinking about areas because so much of our early learning about numbers and measurements is linear.”¹⁶ Perimeter will likely make more sense to students since it is linear and that is what they are well practiced in. “There are no tools in the set of school mathematical instruments to measure area: areas are almost always the result of calculations, and are seldom measured directly.”¹⁷ While there are many manipulatives to help students calculate linear measurements, there are none that specifically measure just area. Area must be calculated with the use of an operation, or by counting. Area of rectangles is typically taught using multiplication. For this reason, it is not often taught in the lower grades.

However, the area of polyominoes is usually best found simply by counting. So I can introduce the idea of area to my students using polyominoes. Then we will graduate to finding the area of rectangles. I plan to teach my students about area of rectangles in terms of repeated addition. This will eventually become multiplication in the higher grades. For example, if students have a 3×3 rectangle decomposed into squares they would have three rows of three and three columns of three. They could either count the total number of squares used to create this figure or add 3+3+3 representing the three rows of three to get an area of nine square tiles as shown in Figure 5.

Figure 5: Students can clearly see that there are three groups of three squares. They would simply count all the squares that compose the shape, or if they are more advanced in their math skills they could add three blue squares plus three yellow squares plus three purples squares to get a total of nine squares.

Area and perimeter can be determined for pentominoes and their enclosing rectangles similarly. Students can count the edges of the pentomino figure to determine the perimeter and can count the total number of square tiles used to create it to determine the area of the figure. They would determine the area and perimeter of the enclosing rectangle when they filled in the gaps in the array structure to compose and complete rectangle and then count the edges for perimeter and total number of tiles used to determine area (as shown in Figure 6). They may be surprised to find that, although the area of the enclosing rectangle is always larger than the area of the pentomino (except for the straight bar, when they are equal, since this shape is a rectangle), the perimeter of the enclosing rectangle is never larger, and can be smaller. Students will need to explore the pentominoes to discover this.

Figure 6: A pentomino was created by using five square tiles. Its enclosing rectangle was created by completing the shape and adding squares to complete the shape to a full rectangle (which in this case is a square). The areas and perimeters have been determined for both shapes. The blue represents the original pentomino and the purple shows the enclosing rectangle.

While these concepts are not required at the first grade level, I am confident that my students will be able to reach them with support and using proper strategies. I also expect that if they begin this work in first grade they will be less likely to have misconceptions when they reach testing grades. In doing this work I believe I will help my students see how geometry and measurement are truly a part of mathematics and set them up to be successful in future years.