Trapezoid Classroom Problem Andrew Bartow Defining the Problem A teacher needs to fit 19 students in her classroom around a set of 9 trapezoidal tables (with a 10th auxiliary table available if needed). Using these 10 trapezoids with given dimensions, what configuration is the most efficient use of the tables and of the room that is able to seat the 19 students? If possible, the teacher wishes for all students to face each other. Defining the Trapezoidal Table and the Classroom Note: All measurements and values are given in inches unless stated otherwise. Figure 1 provides side length measurements of the isosceles trapezoid tables. While these measurements may prove useful, we may later need additional information. For example, we may need to know the height of the trapezoid, specifically, the length of a segment in the interior of the trapezoid perpendicular to the short and long parallel sides. Knowing the measures of the interior angles of the trapezoid may also prove useful when we attempt to model the problem physically and digitally. We can quickly calculate these values using trigonometry. Figure 1: Trapezoidal table with dimensions Calculating the Height The height of the isosceles trapezoid is the line segment contained in the interior of the isosceles trapezoid perpendicular to both parallel sides. Constructing the auxiliary height segment forms a right triangle with the slanted side, the height, and a portion of the long parallel side of the isosceles trapezoid as its sides. By the definition of an isosceles trapezoid, the short parallel side is centered in reference to the long parallel side. Specifically, auxiliary height segments whose endpoints coincide with the short parallel side will create two congruent right triangles with congruent segments contained in the long parallel side. Thus, we can calculate to find how far along the long parallel side the height and that line segment intersect, making the length of that portion of the long parallel side included in the right triangle. We now have 2 side lengths of a right triangle, and using the Pythagorean Theorem can calculate the third, which is the height. Calculating the Angles Using the same right triangle we formed above, we can calculate the measure of the angle formed by the intersection of the slanted and long parallel sides easily using a trigonometric function. The angle formed by the intersection of the short parallel side and the slanted side can be found by adding the other angle in the right triangle to 90 degrees, since the short parallel side and the height of our constructed right triangle form a right angle, and we know the interior angle sum of a triangle is 180 degrees. The measure of the angle of interest is approximately 117 degrees. ( ) Labeling the Trapezoid Figure 2 is simply a graphical representation of the trapezoid made using Geogebra with the information listed above. Figure 2: Measurements of the Isosceles Trapezoid Tables This image also illustrates two pitfalls of Geogebra as a modeling method: it is very easy to distort the shape by a small amount causing slight changes in dimensions that may compound, and there is also round off error in the computed values. Concerning the Size of the Classroom The room measures 18 feet by 15 feet. This is equal to 216 inches by 180 inches. We were given these measurements by the teacher at the start of the investigation. However, a field trip to the actual classroom revealed several concerns. An optimal solution must provide sufficient space around the edges of the table layout for the teacher to move around the classroom. Other objects in the room contribute additional constraints we did not anticipate based on the initial photograph. Finally, the presence and positioning of the dry erase board and BrightLink interactive board strongly suggest all students should be seated such that they could see the long side of the room. Modeling Through Trial and Error Polya claims the first step to solve a problem is to understand the problem. To better understand the problem, I sat down and sketched a few diagrams to try and see relationships between tables on a small scale. Figure 3: Hand drawings modeling the case where n = 2 (where n is number of tables) A few drawings later, a number of things become apparent. The first was that I was definitely not good enough at drawing trapezoids to solve this problem by hand. It would be necessary to use Geogebra as a modeling tool. The second was how the tables fit together. One could alternate the orientation of the tables and make a single straight piece. If one were to place them side by side, there would be a lot of seating on the outside, but that shape appeared unstable. These were just initial observations, but they would help guide later explorations. The next step involved writing out more sophisticated cases and how many students each case would seat. (a) (b) (c) Figure 4: Selected hand drawings for cases with n > 2 Obviously these were crude models. There were still things to be learned from them, though. A small square could seat 8 students with just 4 tables (4b above). If we could expand this configuration, using the 10 available tables, we could seat 20 students. This layout would provide sufficient seating, so the next step was to scale the design up. Figure 4c had 4 sides, each seating 5 students, for a total of 20 students. However, it took 3 tables each to make those 4 sides. We do not have 12 tables available. We appear to lose seating as we scale up the number of tables. At some point, I thought it would be necessary to remove tables from the corners of Figure 4c. This led to drawing Figure 4a. It might not even have worked out in, but with 10 tables it could seat 17 students. The distortion in my hand drawing prevents the reader from gaining an accurate sense of how many students are able to sit around Figure 4a. Figure 4a is an improvement in regards to seating more students with fewer tables. Testing Out a Design (or, Finding a Fatal Flaw) If one simply tries to find a solution quickly, it is very easy to come up with a very bad idea that might showcase things to avoid. There are many different ways to arrange the various trapezoids. One possible solution is shown below. Figure 5: The design I field tested in the actual classroom This shape appears to fit many of the requirements. It can easily seat 19 students with ample room and prevents students from backing into each other while seated. This is the result of the design taking up a lot of space in the classroom and using two large angles where tables intersect in the middle of the classroom. The seating around the two acute angles still isn’t very efficient. The design can be made with less than nine tables, but as students rearrange themselves around the tables, it is likely two adjacent students will back into each other while seated. Students also do not face each other, and it can be hard for many of them to see the whiteboard. This is a useful exercise for seeing potential problems, but without an organized and efficient plan, we may fail to optimize the seating arrangement. In field testing this design, I discovered the initial photograph and problem statement lacked many of the artifacts in the classroom, such as bookshelves, that placed additional constraints on the table layout. Modeling the Problem Rigorously with 2 Trapezoids Working Configurations Figure 6 Figure 7 Seating Capacity: 8 Seating Capacity: 8 Figure 8 Figure 9 Seating Capacity: 8 Seating Capacity: 6 Non-Working Configurations Figure 11 Figure 12 Seating Capacity: 7 Seating Capacity: 6 Figure 13 Seating Capacity: 6 Examining Similarities and Differences to Determine Mathematical Structure The configurations fall into two distinct groups that can be termed “working” and “non-working”. Configurations in the working group tend to be either a convex polygon or include large angle measures. This allows the students seated in these configurations to move in and out of their chairs freely. Upon inspection, the configurations in Figures 11 and 12 suffer from another problem we should avoid: they do not “lock”. Because these configurations have parts of tables jutting out or attached only at a single point, tables in these configurations will likely shift as students move in and out of their chairs and require constant adjustment to maintain formation. Defining Individual Workspace and Chair Freedom If one were to seat two students between the left slanted side of the trapezoid on the bottom and the long parallel side of the trapezoid on the top of Figure 12, one could picture easily the problems that might surface with both students attempting to slide their chairs out of the table at the same time. There simply is not enough space to do this. Students tend to slide their chairs back a fixed length in any direction. The fixed length is about 14 inches for these sort of chairs based on our field test. Thus, we can model the room the needed for free chair movement using a semicircle centered at where the student is seated with a radius of 14 inches. The semicircle intersection responds to the distance between the centers of the two semicircles. So long as the centers are more than 28 inches apart, we will not run into any significant issues. When we are constructing these configurations, we have two ways of influencing the distance between the centers of the two semicircles: we can either place the tables in a line with students sitting far apart or we can angle the tables.