Demystifying Area and Perimeter: A Practitioner’s Guide to Strategies That Work

Strategy Rationale

The first strategy for teaching area and perimeter to students with LD in mathematics is incorporating the use of manipulatives. Manipulatives are used in the mathematics classroom to represent an idea or a process in concrete form and can have benefits for student learning (Bouck & Park, 2018). Manipulatives facilitate learning by helping students make connections between concepts and visual representations. For instance, fraction bars can be used to represent fractional parts and help students make sense of the magnitude of the fraction (Barbieri et al., 2019). As students construct knowledge from the use of manipulatives, they can transfer that learning to abstract symbols. Students with disabilities benefit from instruction that intentionally helps them connect their knowledge, constructed from manipulatives, to abstract symbols in an explicit manner. This is known as the concrete-representational-abstract (CRA) sequence (Bouck et al., 2017; Flores et al., 2014; Strickland & Maccini, 2012). In this sequence, students first learn a mathematics concept with concrete manipulatives, then learn to draw a two-dimensional representation, and finally, solve using only abstract numbers and symbols. An additional benefit to manipulatives is that they have led to improved retention of mathematics skills over time (Cass et al., 2003).

Using Manipulatives in the Classroom

Teachers can use a geoboard, a square board manipulative with pegs arranged in equal rows, to teach area and perimeter. The geoboard has led to improved area and perimeter understanding for students with LD when paired with explicit and systematic instruction (Cass et al., 2003). This strategy can be particularly beneficial when first introducing area or perimeter because the use of manipulatives can aid in concept development. The following scenario models how this can be implemented in the classroom.

Begin by defining perimeter as the length around a shape. Then, provide students with perimeter problems selected from textbooks and have them represent the shape on the geoboard using the same number of units (see Figure 1). Tell students that the geoboard is a representation of the figure in the problem; therefore, they can use the same process of counting around the shape to find the perimeter. Explain to the student how to count the units around the shape in a step-by-step manner. For example,

We will be counting the length around the shape to determine the perimeter. Run your finger along the outside of the rubber band and count each space between the pegs. The total number of spaces, or units, is the perimeter of the object. Watch me as I model this for you. One, two . . .

A common error made by students with LD is to count the pegs instead of the spaces. Provide immediate, corrective feedback if this mistake is made. After spending two or three sessions on perimeter, the teacher can introduce the concept of area. Teaching area with geoboards follows a similar process, however, instead of counting the units around the shape, students will learn how to count the square units on the inside.

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Figure 1.

Using geoboards to teach area and perimeter. The student has used a rubber band and a geoboard to represent the dimensions of the pentagon.

Alternatively, teachers can use virtual manipulatives to teach area and perimeter. Virtual manipulatives are becoming more common mathematics tools because they are easily available, often low cost or free, and can be accessed from the Internet (Bouck & Flanagan, 2010). The Toy Theater ¹ has an area and perimeter explorer that allows students to add square tiles to a virtual grid. Grids are excellent tools for teaching geometric measurement because the units are outlined prior to placing shapes over them, and therefore, act as “area rulers” (Van de Walle et al., 2019). The following scenario will demonstrate how the grid can be used when teaching the formulas for area and perimeter of squares and rectangles.

First, have students create a three by two–unit rectangle with square tiles on a virtual grid. Then say,

To calculate the area of a rectangle, you can multiply the rectangle’s length by its width. In this example, we will multiply 3 times 2, which is 6. This works because there are two rows of three tiles. So, the area of this rectangle is six units.

Have students verify that the answer is correct by counting the square units inside the shape. Next, introduce the formula for calculating the perimeter. Say,

To calculate the perimeter of a rectangle, you can multiply the width by 2 and the length by 2. This works because we have two sides of each measurement length. In this example, we have two sides that are three units long and two sides that are two units long. Let’s multiply 3 times 2 to get the total length and 2 times 2 to get the total width. Then, we will add the total length to the total width to get the perimeter.

Have students count the units of length around the rectangle to verify that their answer is correct. Figure 2 is an example of how to connect the visual representation of the manipulatives with the abstract formulas of area and perimeter.

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Figure 2.

Connecting area and perimeter to the formula. Presenting the formula with the visual representation helps students make the connection between the two. A = area; P = perimeter.

To assess student understanding, the teacher can have them use a geoboard or virtual grid to solve area and perimeter problems independently. Watch students as they count the units to ensure they are counting the border units for perimeter and the inside, square units for area. To increase the rigor of the task, teachers can have students match area and perimeter problems to their corresponding equations.

Strategy Rationale

Discussed next is Strategy 2: focusing on the variable. For many students, the presence of an irrelevant, yet more obvious variable than the one at hand can be a distraction on visual mathematics tasks (Dembo et al., 1997). For instance, when presented with two shapes that have congruent perimeters and incongruent areas, students may respond that the perimeters of the shapes are incongruent because the visual distraction of the incongruent areas interfere with students’ reasoning. However, drawing attention to the variable of interest reduces the effect of the interference of irrelevant variables on students’ thinking (Babai et al., 2016). In this example, the student’s attention should be directed toward the perimeter (i.e., the variable of interest) instead of the area. Teachers can do this by utilizing a discrete mode of presentation, as opposed to one that is continuous. A discrete mode, as demonstrated in Figure 3, is one in which the line segments of the shape are not connected (e.g., using a dotted line or toothpicks to create a rectangle). Using a discrete mode of representation draws the attention toward the perimeter, which has been successful at increasing accuracy on comparison of area and perimeter tasks in which either the area or perimeter differ (Babai et al., 2016). Thus, it can be used in the classroom to dispel the misconception that incongruent perimeters must always have incongruent areas and vice versa.

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Figure 3.

Discrete mode of representation. The discrete mode of representation emphasizes the perimeter of the triangle.

Using Focusing on the Variable in the Classroom

Helping students focus on the variable can be beneficial for students who frequently confuse the concepts of area and perimeter. Teachers can utilize Strategy 2 by displaying two similar shapes in which the perimeters are different and use a discrete mode of representation to draw students’ attention toward the perimeter. First, start by creating a two–unit by four–unit rectangle with individual square units made of cardstock (i.e., two squares in each of four rows would make a 2 × 4 dimension rectangle). Model how to calculate the area by counting the square units inside the shape. Then, cut several pieces of straw that are the same length as the side of the square units made of cardstock. Place the straws around the border of shape, so that the focus of the task is on the perimeter. Model how to calculate the perimeter by counting the straws. Record the area and perimeter of this arrangement on a white board. Remove the straws and rearrange the square units into a different shape, so that the perimeter is different. Again, place the straws around the new shape to draw the focus to the perimeter. Use the same steps as before to calculate the new area and perimeter and record it on the white board. Discuss why the area remained the same and the perimeter increased (i.e., more edges of the square units were exposed).

Likewise, when comparing shapes with the same perimeter and different areas, teachers can increase the focus of the area by coloring in the shapes, as shown in Figure 4. After the shapes are colored, have students draw the columns and then the rows inside each shape to create square units. Then, the students should count the square units inside of each shape to calculate the area. Finally, encourage students to discuss why the areas are different even as the perimeters are the same. Spend two or three sessions on the same problem type (e.g., same area different perimeter) before introducing a different problem type (e.g., same perimeter, different area). Then, spend at least one session in which the student must differentiate between the two problem types before solving.

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Figure 4.

Activity: same perimeter, different area. This activity will help students understand that area and perimeter measure different things.

To assess understanding, teachers can provide students with a worksheet that has a mix of area and perimeter problems. Prior to having the student solve independently, the teacher can draw focus to the variable of interest by drawing discrete units around the shape when it is asking for the perimeter or drawing the square units inside the shape when it is asking for the area.

Strategy Rationale

The third instructional strategy that can be used to support students during area and perimeter problem-solving is the use of VCRs. Using VCRs can help students to retain more visual information about a problem by reducing the cognitive load of the task (Zhang et al., 2012). This strategy has been shown to improve students’ visual working memory, a cognitive process on which students must rely for successful mathematical problem-solving (Zhang et al., 2012). Students with LD often exhibit pronounced deficits in visual working memory (i.e., the ability to create, store, and manipulate visual information), which can make seemingly simple area and perimeter tasks extremely frustrating (Baddeley, 1992; Phillips, 1974; Zhang et al., 2012). Visual chunking can be provided during area and perimeter tasks to bring attention to related elements or distinctive features of a problem by strategically “chunking” information and presenting it to students in a visual manner (Bobis et al., 1993; Zhang et al., 2012).

In Figure 5, an area problem is represented in the following two different ways: (a) using the standard representation and (b) using VCR. Notice the comparison between the standard form and VCR. The standard representation provides students with the following: (a) the shape of the figure, which should cue students as to which formula to use; (b) one side length; and (c) one width. However, it does not explicitly indicate that opposite lengths and widths are equivalent, which is necessary prerequisite knowledge for solving the problem. Using the standard representation, students must retain at least five pieces of information: (a) the shape type, (b) the shape’s attributes, (c) the given length, (d) the given width, and (e) relevant knowledge for determining the appropriate formula to use. In contrast, the VCR utilizes visual chunks to cue to students to the important visual features within the representation. The act of chunking makes it visually apparent to students that there are three groups of six units inside the rectangle by shading and partitioning the rectangle into individual square units. In addition, the length and width are labeled with numbers and corresponding units, along with clearly marked hash mark symbols to represent that opposite sides of the figure are equal. Using VCR, the student can recognize this representation as a “single informational chunk” that contains information about the figure’s attributes and cues the retrieval of relevant semantic knowledge, such as the steps to solve for the area or perimeter (Stieff et al., 2020, p. 3).

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Figure 5.

Standard representation versus visual-chunking representation. This figure shows an example of the standard representation of an area problem and the example of an adaptation using VCR.

Using Visual-Chunking Representations in the Classroom

Incorporating VCRs into daily instruction requires making adaptations to existing materials (i.e., textbook examples or curriculum guides). VCRs can be used at any point in the lesson by replacing the standard representations with the adapted VCR. Students will receive the most benefit from VCRs when they (a) have prior knowledge of area and perimeter, (b) have had adequate exposure to representations, and (c) can utilize the visual supports with minimal support from the teacher (e.g., during independent practice opportunities and exit tickets). To teach students to use VCRs during problem-solving, begin by reviewing key concepts related to area and perimeter problems. Ask students to turn-and-talk with their partner about what they know about area and perimeter, before asking them to share with the whole group. This gives the teacher an opportunity to circulate and monitor student discussions to informally assess understanding and determine the level of scaffolding needed during subsequent parts of the lesson. It also gives an opportunity to address misconceptions, clarify concepts, or provide corrective feedback. Next, explicitly review previously taught vocabulary terms that are relevant to the lesson (e.g., area, perimeter, parallel). Pairing area and perimeter strategies with vocabulary instruction is critical for students to access course content, deepen comprehension of concepts, and improve mathematical proficiency (Riccomini et al., 2015).

After reviewing vocabulary, introduce students to VCRs by showing students an example of a VCR that was adapted from a standard textbook problem (e.g., the one shown in Figure 5). Say, “This is a visual-chunking representation, but we can call it VCR. A representation is a picture that gives us information that we can use when we are solving area and perimeter problems.” The next step is to model thinking aloud while emphasizing key features of the figure, along with any included symbols (e.g., hash marks to represent equivalency). Begin thinking aloud by identifying the information explicitly given in the representation (i.e., length and width). Point to the figure while describing its features and have students do the same. For example, point to each set of sides, noting the length and width and say,

I know this is a rectangle because it has two sets of opposite sides that are equal and parallel. I’m given the length of this side, 6 inches, and the width of this side, 3 inches. I also know that this symbol (a single hash mark) means that the opposite side length is equivalent, so it is also 6 inches long. The double hashmark on opposite sides of the figure also tells me that the opposite widths are equivalent. If one width is 3 inches, the missing width must also be 3 inches.

Then, the teacher can talk students through the process of determining which formula to use and model how the representation can help them to check their work for accuracy. For example, say, “For an area problem, I know I can use the formula ‘area equals length multiplied by the width’ (i.e., A = l x w). I could also double check that I performed the correct calculation of area by counting each square unit.” Answer any questions that may arise before continuing to guide students through additional guided practice problems; gradually releasing the responsibility to the students to prepare them for independent practice. Repeated practice will increase students’ sensitivity to visually chunked information and enable students to more efficiently utilize chunked information during problem-solving tasks (Stieff et al., 2020). After completing two or three practice problems together, the teacher can assess students’ ability to use VCR unprompted by having them complete the remaining practice problems on their own. During this time, the teacher monitors performance and provides corrective feedback (Archer & Hughes, 2011). Finally, the answers are reviewed as a whole group, followed by a brief recap of key points, before releasing students to complete four or five independent practice problems. Be sure to choose problems that are similar to those modeled during the lesson.

Strategy Rationale

The fourth strategy, contextualized instruction, incorporates real-world scenarios and experiences into mathematics instruction. Effective mathematics lessons are those that offer opportunities for students to develop an understanding of mathematical ideas through the incorporation of relevant, familiar, and interesting contexts for problem-solving (Van de Walle et al., 2019). Concepts of geometric measurement, especially area and perimeter, easily can be connected to real-world situations; however, for students with learning disabilities, it can be difficult to understand the relevance of applications of mathematics concepts outside of the mathematics classroom (Espejo & Deters, 2011). As evidenced by state standards, students are expected to apply learned mathematical knowledge to complex, novel problems across environments, settings, and contexts. The use of relevant real-world problems helps students to create meaningful connections between school-based and real-world applications of mathematical concepts (Van de Walle et al., 2019).

Using Contextualized Instruction in the Classroom

Teachers can provide opportunities for contextualized instruction in a multitude of creative ways that increase motivation for task completion and lead to greater skill maintenance and generalization over time (Collins et al., 2011; Van de Walle et al., 2019). It can be used effectively throughout the lesson progression and across grade levels. A simple but effective way to do this is to incorporate relevant scenarios into existing area and perimeter tasks. For example, imagine a problem that asks students to solve for the perimeter of a rectangle that has a length of 9 inches and a width of 4 inches. Although the task is straightforward, generalization may not occur if students do not understand the rationale for calculating perimeter or its significance in the real world. To adapt the problem, simply insert information that students will connect with, such as names, personal interests, or cultural traditions. An adapted version of this problem is shown in Figure 6. Note the adjustments that were made to make the problem more meaningful: (a) the teacher’s and students’ names are included, (b) there are references to students’ cultural traditions, and (c) students are being asked to apply measurement concepts in context. When students recognize scenarios within problems that are relevant to their lives, the creation of mental representations associated with the problem becomes less difficult, leaving cognitive resources available for determining the correct problem-solving path (Satsangi et al., 2019).

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Figure 6.

Word problem-solving using contextualized examples. This is an example of how to integrate students’ relevant personal and cultural experiences into word problems.

Contextualized examples can be used throughout the lesson progression, including during guided practice, independent practice, exit tickets, and assessments (e.g., teacher-made unit tests). To assess students’ ability to use the strategy independently, provide students with three or four contextualized problems to complete on their own. Collecting assessment data is critical for planning appropriately differentiated instruction. For example, more advanced learners can be tasked with generating their own single or multi-step word problems. For struggling learners, it would be more appropriate to utilize teacher-created problems and provide scaffolding support until they become more confident problem-solvers (Van de Walle et al., 2019). Regardless of skill level, word problems can be tailored to reflect student interests and experience, while also being appropriately aligned with students’ individual needs and the academic course content.