In this section, we feature teacher reflections related to challenging to teach mathematics topics . These reflections include some common student difficulties and errors as well as ways of addressing these challenges If you have such a topic you wish to share with other practitioners in the field, please respond to the call for manuscripts.
Getting it Right with Trapezoids
Dr. Agida Manizade and Dr. Marguerite Mason
The area of a trapezoid is one of the topics commonly taught at the middle school and high school levels of mathematics (NCTM, 2000). When students are given the formula to memorize, they do not get an opportunity to develop their conceptual understanding of the area of a trapezoid. It takes time to allow students to explore and derive the formula on their own ; however, this time is well-spent, as students develop their own deep understanding of the mathematical concept (Manizade and Mason 2014).
Typically, the area of the trapezoid is introduced after students have had an opportunity to learn about the areas of a rectangle, a parallelogram, and a triangle. Given that they already understand and know the aforementioned concepts, here is how I typically approach this topic with my students. I start the conversation by focusing on the definition of a trapezoid, and make a note that there are two existing definitions of this mathematical construct. The Virginia Department of Education (VDOE) defines trapezoids as a quadrilateral with exactly one pair of parallel sides. An alternative definition of this concept is more inclusive; a trapezoid is a quadrilateral with at least one pair of parallel sides. According to this definition, shapes such as squares, parallelograms and rectangles are special cases of trapezoids. In my classroom, I often include a discussion about the differences between the two definitions and find it to be mathematically engaging for students. For the purpose of this article, we will be using the first definition, as this is the one used on the Virginia Standards of Learning.
Fig. 1 A decomposing and recomposing approach using triangles.
Posing the Problem
I start by asking students to sketch a trapezoid. Then, I ask students to consider formulas for the area of a rectangle, a parallelogram, and a triangle, and think of the ways to derive the area of a trapezoid using these formulas. We discuss their solutions as a class. If no one has offered a decomposing and recomposing solution, I present the following solution to them. See Figure 1. As a whole class, we discuss how to derive the formula for the area of a trapezoid using this approach, given that we know the area of a triangle. I take time to compare the heights of each of two triangles in Figure 1, so that students understand that a trapezoid can be decomposed into two triangles with equal heights and the bases of these triangles are the bases of a trapezoid. We then complete a formal geometric proof using the outlined approach. Once students see a solution presented to them, I then ask them to develop their own strategy using their knowledge of areas of a triangle, rectangle, or parallelogram. Below we describe a set of strategies, some limited and some generalizable that are commonly used by students.
Fig. 2 A special case of an Isosceles Trapezoid.
We do not provide formal proofs in this article, as our goal is to present you with analysis of the most common difficulties related to developing the conjectures regarding the area of a trapezoid.
Different Approaches for Deriving the Formula
There are three types of approaches students typically use when deriving formula of areas of a trapezoid (Manizade and Mason, 2014). The Decomposing and Recomposing Approach is when a student takes a case of a trapezoid and decomposes it into shapes such as rectangles, parallelograms, and triangles. Then, students calculate individual areas of the shapes and sum them up to derive the area of a trapezoid. In some cases, they combine two subdivided pieces and find the area of a shape after which they add it to the remaining parts, as seen Figures 1 and 3.
Fig. 3 A generalizable decomposing and recomposing approach.
The Transformational Approach occurs when students use transformational geometry to move parts of the trapezoid in order to create a triangle, rectangle or parallelogram, as presented in Figures 4 and 5.
Fig. 4 Transformational approach using a triangle.
Fig. 5 A transformational approach using a parallelogram.
Finally, the Enclosure Approach, happens when a student encloses the shape of a trapezoid into another shape, such as a rectangle, parallelogram, or a triangle.
Fig. 6 An enclosure approach using a triangle.
The student then calculates the area of the newly created object, and subtracts the portions that are not part of the original trapezoid, as presented in Figures 6 and 7.
Fig. 7 An enclosure approach using a parallelogram.
Any one of these three approaches is appropriate and mathematically valid. Students’ solutions range widely, depending on their background, their algebraic skills and their abilities to construct formal proofs. Some of these strategies involve very simple algebraic procedures, where others are more complex. In my experience, regardless of the student’s background, given the opportunity, each student is able to come up with at least one mathematically valid generalizable approach for deriving the formula for the area of a trapezoid. Depending on their ability to construct geometric proofs, they may or may not be able to formulate the outlined strategy. In my geometry class, my goal is for students to be able to identify multiple strategies for deriving the area of a trapezoid formula, and provide a formal proof corresponding to each strategy. Naturally, not all solutions presented by students are generalizable for all trapezoids, and students often use special cases to derive the formula.
Commonly Observed Challenges and the Causes
In my classroom, students typically spend between 15 and 20 minutes developing their own strategies. Each time I conduct this activity, I observe the same set of common errors and difficulties described below.
Fig. 8 A special case of a Right Trapezoid
Challenge 1: Students use special cases of trapezoids. For example, in Figure 2 students are using an Isosceles Trapezoid.
They typically decompose Isosceles Trapezoids into two, congruent right triangles and a rectangle or a square. Then they calculate the areas of each shape and sum them up to derive a formula for the area of a trapezoid. The reason students make the errors mentioned above is related to student’s understanding of trapezoids. If a learner only thinks of a trapezoid as an Isosceles Trapezoid, and does not recognize that there might be trapezoids that do not have two equal sides, they are likely to present a limited strategy for developing the area of a trapezoid formula, such as in Figure 2.
In addition, some students think that the rectangle formed in the middle is always a square, which leads to an incorrect formula during the formal derivation process. This challenge is typical for students at a Van Hiele Level 1: Visualization, and is most common for students who use graph paper to do their sketches and calculate the area by counting the number of squares rather than using algebraic reasoning to derive the formula.
Challenge 2: When using a special case of a right trapezoid as seen in Figure 8, the students decompose a trapezoid into a right triangle and a rectangle, and find the area of a trapezoid by adding both areas.
If a student thinks of a trapezoid only as a Right Trapezoid, then they are likely to produce a limited strategy for deriving the area of a trapezoid formula, as presented in Figure 8.
Challenge 3: Often, students produce a strategy where a non-Isosceles Trapezoid as shown in Figure 9 is used.
Fig. 9 A special case of a non-Isosceles Trapezoid approach part 1.
Fig. 10 A special case of a non-Isosceles Trapezoid approach part 2.
In this approach, students create two right triangles with the same height and combine them to calculate the area of the newly developed triangle as seen in Figure 10. Then, they add the area of the triangle to the area of the remaining rectangle. This outlined strategy does produce an accurate formula for the area of a trapezoid however; it is also a special case as it does not work for all types of trapezoids. This approach will not be appropriate for a trapezoid which has bases that do not form an enclosed rectangle, as shown in Figure 11.
Fig 11. Trapezoid ABCD
If a learner only thinks of a trapezoid as a quadrilateral, where the bases are located directly above each other and does not think of a trapezoid presented in Figure 11, they are likely to present this limited strategy for developing a formula for the area of a trapezoid.
Fig. 12 Special case of a trapezoid transformed to a rectangle.
Challenge 4: Another commonly presented special case is when students only consider an Isosceles Trapezoid, then decompose it into a right triangle and a right trapezoid, then recomposes it into a rectangle, as shown in Figure 12. The student then derives the area of a trapezoid by finding the area of a rectangle, AFCE, which is equal to the area of the original trapezoid, ABCE (see fig 12).
Similar to Challenge 1, in this case, the student typically thinks of a trapezoid as an Isosceles Trapezoid and does not recognize there might be trapezoids that do not have two equal sides.
Student Misconceptions Underlying the Challenges
To address these student challenges, I recommend spending some time to fully develop the construct definition for the trapezoid by: 1) presenting samples of examples and non-examples of trapezoids, 2) critically analyzing the formal definition of the trapezoid, and 3) asking students to sketch as many different types of trapezoids as they can. This can eliminate the aforementioned student challenges related to limited understanding of the concept of the trapezoid.
After modeling an example for your entire class, allow students time to develop and test their conjectures and strategies as well as encourage them to test those for other types of trapezoids. Providing students with graph paper and construction tools to test their conjectures as they develop them, and using dynamic software where students can calculate the areas to test their conjectures before engaging in a formal proof is also useful. Once students realize the limitations of their strategies, I then prompt them to consider another strategy that is generalizable for all types of trapezoids, not just special cases. After students had the opportunity to develop their strategies and/or formal geometric proofs, I allow them to share, compare and contrast their approaches with the rest of the class, and solicit critical feedback from their peers. Doing so can provide a rich and meaningful mathematical discourse by giving students the opportunity to engage in mathematical reasoning on a deeper level.
This is one of the tasks that have the potential to engage students in a higher-level of thinking in a middle school or high school geometry classroom. It is appropriate for diverse students with different mathematical backgrounds and skill sets. In my experience, once students had an opportunity to explore the area of a trapezoid in this manner, not only were they able to recall the formula, they were also empowered by knowing multiple strategies of deriving it. This is more important than being able to implement a memorized formula. Within one task, a teacher can increase student’s geometric development from level one to level four of van Hiele’s Levels of Geometric Development (Mason, 1998). This can be achieved by providing follow-up, leading questions during the small group discussions and whole class discussions. This type of activity can also be used as a formative assessment to gather information about student’s geometric level of development. Within the activity, the teacher has an opportunity to then differentiate the instruction and address the needs of each student. In my view, taking the time to discuss and critically evaluate special cases presented by students and identifying their limitations can generate a valuable mathematical discussion at a deep level.
Mason, Marguerite M. 1998. “The van Hiele Levels of Geometric Understanding.” Professional Handbook for Teachers: Geometry, edited by L. McDougal, pp. 4-8. Boston: McDougal- Littell/Houghton-Mifflin.
Manizade. A. G & Mason, M. 2014. Developing the area of a trapezoid. Mathematics Teacher 107 (7). 508-514.