Michael S. Meagher, Michael Todd Edwards, and S. Asli Özgün-Koca The Geoboard Triangle Quest yields many results. The challenge for students is to verify which—if any—are correct. In her Mathematics Teacher article “The Geoboard Triangle Quest” (2013), Kasi Allen describes students’ excitement as they explored how many triangles can be constructed on a 4 × 4 geoboard. Because the task engages students in many Common Core Standards for Mathematical Practice (CCSSI 2010)—for example, “Look for and make use of structure” (SMP 3) and “Model with mathematics” (SMP 4)—and is particularly well suited for multiple solution strategies, we posed the task to our preservice teachers (hereafter referred to as students) as a problem of the week. We posed the Geoboard Triangle Quest in this form: How many noncongruent triangles can be constructed on a 4 × 4 geoboard? How do you know? Justify your answer with significant supporting work. The use of advanced digital technologies as tools for problem solving receives much attention in our methods classes, and we anticipated that some students would use technology to solve the task. We looked forward to the interplay between those approaches and hands-on approaches mirroring those adopted by Allen’s students. Ultimately, our students attempted the task in many ways, using both digital and analog techniques. Interestingly, different solution strategies yielded different numbers of triangles. Although Allen’s students tentatively concluded that 72 triangles were possible (2013, p. 118), they did not provide a proof, and none of our students arrived at this answer. In this article, we discuss how we came to what we consider a definitive solution and, on the way, address a challenge that arises in many inquiry-based classrooms: When confronted with multiple, conflicting results, how does one determine which—if any—answers are correct? INITIAL SOLUTION METHODS Hands-on with Geoboards A number of our students used physical geoboards to explore the task, taking approaches that paralleled those discussed by Allen (2013). Students found it easy to construct arbitrary triangles with rubber bands. Soon, however, it became clear that a more systematic approach would be needed. Recognizing that fourteen distinct segment lengths can be constructed—for example, —several groups began by forming all triangles containing a shortest side of length 1. Next, they constructed all triangles containing a shortest side of length They continued, constructing triangles with shortest side lengths 2, then then and so forth. Constructing an initial segment with desired length, students stretched the band to a third peg to form a triangle with no sides shorter than the first. We illustrate this approach in figure 1 for initial length . Table 1 summarizes the initial work of one group who used this method. Note that the last four rows of the table do not indicate that triangles with side lengths 5, and are not constructible. Rather, those lengths are not possible as shortest sides. In all, students uncovered 71 triangles with geoboards. Virtual Geoboard Applet A number of students preferred to use GeoGebra to create a virtual geoboard. First, they constructed a digital version of a hands-on board, ultimately modifying the applet to extend its functionality with measurement and data collection features. An online version of the students’ final applet is available at http://bit.ly/triangle-sketch. Interestingly, even the earliest versions of the applet provided students with advantages over a conventional board. For instance, the fixed size of rubber bands made the construction of triangles with small interior angles difficult. Two such triangles, discovered with the applet but initially missed by physical geoboard users, are illustrated in figure 2. Initially, students used the virtual applet as little more than a conventional geoboard—a digital analog that allowed them to construct triangles without the constraints of physical rubber bands. Soon, however, they revised the sketch to make better use of the many capabilities of GeoGebra. An early revision added numeric labels to each side of the triangle. This feature helped students compare side lengths of newly constructed triangles with those already recorded. In a later revision, the students used spreadsheet and scripting features to automate the process of recording triangles. Specifically, a Record to Spreadsheet button was added that copied side lengths and perimeter to a spreadsheet. A second button, Clear Spreadsheet, was added. These buttons are illustrated in figure 3. To help avoid duplicate triangles, the revised applet sorts collected lengths in ascending order before recording the measurements into columns A, B, and C. A final revision added dynamic text indicating the number of distinct triangles currently recorded in the spreadsheet. Controlled by a short script comparing the perimeter of the most recent triangle with all other values in column D, the script adds one to the Unique Triangle count when the perimeter is a new one. This functionality is highlighted in figure 4. Using the final revised sketch, the students systematically constructed and recorded triangles. As figure 5 suggests, 215 triangles were recorded with the applet, 77 of which had distinct perimeters and thus were considered unique. OVERCOMING THE LIMITATIONS OF VIRTUAL AND PHYSICAL BOARDS Despite the options afforded by the virtual geoboard applet, a significant limitation is inherent to the use of boards, both physical and virtual: How does one organize the generation of triangles in a systematic manner so that all triangles are accounted for? Alternative approaches, including the writing of computer code, may overcome this limitation. We will explore one such approach. Scilab Programming Using either a hands-on geoboard or a virtual applet, students risk omitting triangles. In an alternative approach, we begin with every possible configuration of three pegs on the geoboard and remove those configurations that do not yield triangles. Modeling this approach for our students, we wrote such code in Scilab®. A pseudocode of our program is provided in the appendix. A functional Scilab program that implements the methods instructors’ pseudocode is also provided. Our initial Scilab program was short, consisting of eighteen lines of code; however, it makes use of significant secondary school-level mathematics, including the distance formula, floor and modulo functions, and elementary counting (e.g., 25 choose 3). As figure 6 suggests, an initial run of the code returned a list of 80 triangles. WHO IS CORRECT? TRIANGULATING RESULTS At an end-of-week class meeting, the students and their instructor shared initial solutions to the Geoboard Triangle Quest, presenting three different solution strategies. Students who took a hands-on approach with physical geoboards found 71 triangles. Applet users found 77 triangles. Scilab users uncovered 80 triangles. Which, if any, number of triangles was correct? As the class carefully reviewed these findings, sharing lists of triangles across groups, it was apparent that each approach included errors or limitations. Revising Virtual Geoboard Findings Our students’ virtual geoboard made the faulty assumption that triangles with equal perimeters were congruent. When students’ analyzed their data more closely, they uncovered two pairs of noncongruent triangles with equal perimeters, thus revising their total from 77 to 79. The pairs shown in figure 7 led to the inclusion of two additional triangles to the students’ list. With these triangles, the virtual geoboard students had 79 triangles in all.Revising the Scilab Program A spreadsheet aided the instructors in comparing the results generated by their code with the list of triangles generated by the virtual applet. The results were identical, except for an extra triangle with side lengths 1.414235, 2.8284271, and 4.2426406 (i.e., , , and ). Much to our surprise, these side lengths fail the triangle inequality because + = As a result of round-off error, the code as written had failed to eliminate this combination. After it was thrown out, the Scilab-generated list and the virtual applet list were identical—each listing the same 79 triangles. A Collaborative Solution When we compared the 71 hands-on geoboard triangles to the 79 generated using GeoGebra and Scilab, we discovered that the that the following triangles had been overlooked: After verifying that each combination could be constructed on a conventional geoboard, we were confident that Scilab had provided a proof by exhaustion and that exactly 79 triangles could be constructed on a 4 × 4 geoboard. Through additional research, we found that this conclusion was supported by a number of Mathematics Teacher articles (Moser , p. 603; Kennedy , p. 82; Willcut , p. 614), although none of these articles provided a proof of the result. CONCLUDING THOUGHTS The three approaches highlighted in this article demonstrate different aspects of two Standards for Mathematical Practice (CCSSI 2010)—“Look for and make use of structure” and “Model with mathematics,” each highlighting advantages and disadvantages of different solving options. For instance, the hands-on approach with physical geoboards provided students with a kinesthetic model to start visualizing different triangles and begin thinking about noncongruent triangles. We can look to advanced digital technologies as an amplifier (Sherman 2014) to this initial approach, and we see in the use of the GeoGebra applet that the technology offers many affordances. The examples are easier to generate, and we do not face logistical issues with the physical constraints of the rubber bands. Moreover, the spreadsheet capability of the applet allows for organization of the data and also allows the user to check whether new cases are indeed new. However, this “bottom up” approach has the inherent challenge of any proof by exhaustion approach when there is not a clear and systematic way of working through cases: How do you know when you’re done? As we see above and in Allen’s (2013) article, even with a high level of organization and a well-thought-through system, the students came up short with the constructive approach. An alternative is a “top down” approach, in which the challenge of knowing when you are done is replaced by the challenge of deciding which of all possible scenarios are not true scenarios for the situation. The Scilab program works as both an amplifier (Sherman 2014) in that the program first “blindly” generates every possible choice of three points on the board, something quite difficult to do by hand. However, the amplification goes further in that the program, like the applet, can be used to test results and organize the data. If done correctly, this approach leads to the correct answer, although, again, in the case of the top-down approach, the limitations of the technology at first led to extraneous answers. The bottom-up and top-down approach contrast nicely: The latter would appear to work smoothly, but it turned out that having the list of possible triangles generated through the hard work of the bottom-up approach was necessary for our students to understand the limitations of the technology while taking advantage of the affordances. Using multiple approaches for modeling problems is useful not only for the insight offered by different perspectives but also, as we see in this case, for the opportunity for each approach to work as a corrective and verifier of the other approach, each honing the other in reaching a correct solution. Ultimately, in this case, a proof by exhaustion using advanced digital technologies settled the problem. For our students, aside from thinking about proof, the most valuable lesson was seeing how multiple approaches to solving problems can be much more than just different ways to find an answer. Moreover, the commingling and influence of different solutions on each other drives each approach toward a correct answer. ACKNOWLEDGMENTS The authors wish to extend special thanks to Kasi Allen for inspiring them to explore the Geoboard Triangle Quest in greater detail. In addition, we wish to thank Lisa Block of Mariemont High School in Cincinnati for her suggestions regarding ways to extend Allen’s initial work. REFERENCES Allen, Kasi. 2013. “The Geoboard Triangle Quest.” Mathematics Teacher 107 (2): 113–18. Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdfKennedy, Joe. 1993. “Problem Solving on Geoboards.” Mathematics Teacher 86 (1): 82. Moser, James. 1985. “How Many Triangles?” Mathematics Teacher 78 (8): 598–604. Sherman, Milan. 2014. “The Role of Technology in Supporting Students’ Mathematical Thinking: Extending the Metaphors of Amplifier and Reorganizer.” Contemporary Issues in Technology and Teacher Education 14 (3): 220–46. Willcutt, Bob. 1985. “Triangles on a Grid.” Mathematics Teacher 78 (8): 608–14. MICHAEL S. MEAGHER, email@example.com, works at Brooklyn College–CUNY preparing teacher candidates to work in New York City public schools. MICHAEL TODD EDWARDS, firstname.lastname@example.org, is a faculty member in the Department of Teacher Education at Miami University in Oxford, Ohio. He enjoys exploring mathematics and pedagogical matters with his three school-aged children. S. ASLI ÖZGÜN-KOCA, email@example.com, teaches secondary school mathematics education courses at Wayne State University in Detroit. She is interested in the use of technology in mathematics instruction.