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

Array Structure of a Rectangle

I wanted to add an aspect of geometry to my unit since geometry can be clearly connected to arithmetic through measurement ¹². Geometry, arithmetic, and measurement all play a role at the first grade mathematics level but sometimes feel disjointed from one another. Our shape and measurement units often feel out of place in the rest of our mathematics curriculum. I often struggle to create connections to bring these concepts together for my students. Using the array structure of a rectangle can help my students see the relationship between geometry, measurement, and arithmetic.

“An arrangement of objects, pictures, or numbers in columns and rows is called an array. Arrays are useful representations of multiplication concepts.”¹³An array is a mathematical drawing that plots out a space, commonly using a grid structure. Arrays can be drawn in many ways. I will be drawing them as rectangles composed of squares. The reason for this is that squares and rectangles are simple shapes and it is easy to find their measurements. The larger rectangle can be decomposed into smaller squares. This relates to measurement since students can count the number of squares that make up the rectangle. This can be done in the same way they count how many smaller non-standard units make up a larger object. For example, in Figure 1, the rectangle can be shown as a rectangle or it can be decomposed into six squares.

Figure 1:

In order for students to see the array structure of rectangles they will need to use square tile manipulatives. Using these square manipulatives students will be able to construct rectangles on a plane. This is the first step for students to see that they used squares to compose a rectangle. Once students have mastered composing rectangles with squares, they can explore polyominoes. Polyominoes are “plane geometric figure formed by joining one or more equal squares edge to edge.”¹⁴ Polyominoes can be created with any number of squares. In order to keep my students within boundaries that make sense for their young age, I will have them focus on using five squares to create pentominoes.

Pentominoes and Enclosing Rectangles

A pentomino is created when five square tiles are joined with edges touching to create a geometric figure. There are twelve ways to create pentominoes, as shown in Figure 2.

Figure 2: Figure 2 shows the 12 different pentominoes. They are color coordinated based on their enclosing rectangle (the smallest rectangle they will fit inside of). The blue rectangle is its own 5×1 enclosing rectangle. The green pentominoes fit in a 2×4 rectangle. The purple pentominoes fit in a 2×3 rectangle. And the pink pentominoes fit in a 3×3 rectangle, but no smaller one.

Pentominoes mostly have a perimeter of 12, except one of the 2×3 shapes which has a perimeter of 10. Pentominoes always have an area of five tiles. However, the area of the enclosing rectangle depends on the shape. From the above description, we see that the enclosing rectangle can have an area of 5, 6, 8, or 9, depending on the shape of the pentomino. 

Enclosing Rectangles

In order to relate pentominoes back to the array structure of the rectangle, I will introduce my students to the idea of an enclosing rectangle. As mentioned above an enclosing rectangle refers to the smallest rectangle that would cover the entire shape of the pentomino. In order to create such a rectangle, one must “fill-in” the missing square tiles to create the overall rectangle. This new rectangle would cover the original pentomino figure. The relationship between the pentominoes and their enclosing rectangle further establish that square tiles can be used to compose a full rectangle. Or the reversal of this, that a full rectangle can be decomposed with square tiles. Enclosing rectangles can be seen in Figure 3. For example the “plus sign” shape has an enclosing rectangle of 3 high and 3 wide. However, the “P” shape next to it has an enclosing rectangle of 3 high and only 2 wide. While they are both pentominoes, they have different enclosing rectangles. The “Z” shape and the “C” shape provide further examples of 3×3 and 2×3.

In Figure 3, the original pentominoes are shown using blue squares. The enclosing rectangle is shown by completing the rectangle with magenta squares. By completing the rectangles students can see that the rectangles have been composed by squares and can eventually use this to determine area and perimeter.

Figure 3: