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. If we place the students far apart, we are not using our space very efficiently. Angling the tables is more efficient, but it is also trickier. Utilizing large angles or convex figures wherever possible is the best use of our limited space.

Good Bad

Figure 14: Modeling student movement with semicircles Students sitting at the tables need ample room to read and write. We know from our field test that an open textbook placed next to a piece of paper measures 26 inches by 10 inches. Thus, a rectangle with dimensions 30 inches by 13 inches should offer ample room for the student to work as well as some personal space. We will position this rectangle with the midpoint of the length where the student is sitting. Due to the size of the trapezoidal tables, combined with the fact that we are generally not seating more than 3 students at a table, this should not become a substantial issue.

Investigating Generalizability to Larger Cases of n

As we move beyond two trapezoids, we need to consider the tradeoffs we encounter as we add a trapezoid onto the configuration. This can be illustrated well by taking another look at Figures 6, 8, and

9. Whenever a trapezoid is added to a configuration, we will be sacrificing a side that could be used for seating in an effort to extend the overall perimeter such that we end up fitting more students. This is most obvious in Figure 9. In Figure 6, one can see how the addition of a trapezoid can affect the shape of the overall figure by adding in an angle. In these cases, we must consider carefully the angle that we create as to not cause issues with seats running into each other. If the desired result is to simply extend a side, a configuration similar to Figure 8 generates a parallelogram structure. Modeling the Problem with 3 and 4 Trapezoids (n=3 or n=4)

Figure 15: Models for cases where n = 3 and n = 4

Starting at top left, going down the column: Figure 15a, 15b, 15c; starting at the top right, going down

the column: Figure 15d, 15e, 15f

A few interesting patterns emerge from looking at these configurations. Figures 15a and 15d illustrate how we can use the additional trapezoids to extend out angled shapes and how this setup can create sharp angles (it should also be noted that two students would not likely be able to sit on the inside of Figure 15d). Figure 15b indicates how we can make tables into a large block, which provides sufficient space and prevents any seating issues, but fails in that this configuration only seats 8 students with 4 trapezoidal tables and takes up a significant amount of space in the classroom. The bottom two figures demonstrate how adding a trapezoid onto a linear string of trapezoids will cause it to form a long parallelogram when there are an even number of trapezoids, and a long trapezoid when there are an odd number of trapezoids. Figure 15e offers the appeal of a central student focus or line of sight; we will study this configuration in more detail shortly.

Selecting the Shape of a Preferred Configuration

There exists a large number of possible configurations of trapezoids for this problem. It would be impractical to consider even a small fraction of them one by one through trial and error. Thus, we need to narrow down the number of possible shapes of figures we are considering. As discussed earlier, the shape of the mass of trapezoids we are after should be such that all students can see each other and face the same general direction. There should be no small angles around which we seat students. The positioning of the BrightLink in the room, specifically the fact the BrightLink board is parallel to the length of the classroom, suggests the trapezoidal table configuration shape should be longer than it is wide. All of this together points to the need for an elongated semi-circular shape. This shape would not have any sharp angles for students to sit around, allowing the students to see each other as well as the board with a slight adjustment.

Modeling an Elongated Semi-Circle

A simple semi-circular shape was produced during the modeling with 4 trapezoids. We return to this configuration to study it further.

Figure 17: Configuration from Figure 15e revisited

In its current state, this configuration can seat around 10 students. As we are attempting to seat 20 students, we should form 2 of these shapes. After doing so, we are able to seat fewer students, as we are combining trapezoids and losing seating where they touch. However, we can indeed seat enough students in the room by adding a table in the otherwise wasted space in the center.

Figure 18: The optimal configuration of tables in the teacher’s classroom This configuration of tables presents the optimal classroom layout. The requirements as they are stated automatically require trade-offs, as the students cannot sit such that all students face each other and the BrightLink board at the same time. This configuration not only allows us to meet many of the requirements with minimal sacrifice. It allows the teacher to easily change where students sit from class to class as the requirements change. For instance, if the teacher were to have a reading class and wanted all students to face each other, students could be required to sit on the inside of the ring. The teacher can also seat students in the middle table for a writing class that makes use of the BrightLink system. Due to the shape’s large angles, an individual would be able to seat students nearly anywhere without worrying about chairs backing into each other (see Figure 18 above).

Presenting The Case Against the Tenth Table

The tenth table offered in the problem at first seems very enticing. It is difficult to construct an elongated semicircle using ten tables and still fit in the classroom with ample space to move around the room. Adding a table will either increase the shape’s width or length, and we simply lack the space needed to do so. As this problem can be solved without the tenth table and still have more than ample seating, this suggests the tenth table is simply unnecessary.

In Conclusion

It is indeed possible to fit 19 students in an 18 by 15 room, using 9 trapezoids. A set of 2 semi-circles forming an elongated semi-circle is the best way to accomplish this goal. This is shown by small scale testing, considering the requirements for desk space and room to get up from the chairs. My analysis made substantial gains by integration of things learned from trial and error, such as the presence of the

BrightLink board and other objects in the room requiring substantial free space. An elongated semi- circle, shown in Figure 18, provides the optimal classroom setup under the conditions provided by the teacher.

Special Thanks To

This is my first work of mathematics to be published, and I have several people to thank for allowing me this opportunity. I would like to extend thanks to the University of Nebraska-Lincoln, Lindsay Augustyn and Michelle Homp in particular, for allowing me to publish this paper. I would also like to thank my mathematics teacher Mr. Shelby Aaberg for presenting this problem to me and his assistance in readying this paper to be published. The team behind the free software Geogebra I used in most of my modeling in the creation of this solution also deserves a thank you.