aa r X i v : . [ m a t h . HO ] J u l STUDENT INQUIRY AND THE RASCAL TRIANGLE
PHILIP K. HOTCHKISS
Abstract.
Those of us who teach Mathematics for Liberal Arts (MLA) coursesoften underestimate the mathematical abilities of the students enrolled in ourcourses. Despite the fact that many of these students suffer from math anxietyand will admit to hating mathematics, when we give them space to exploremathematics and bring their existing knowledge to the problem, they can makesome amazing mathematical discoveries. Inquiry-based learning (IBL) is per-fect structure to provide these type of opportunities. In this paper, we willexamine one inquiry-based investigation in which MLA students were giventhe space to look for patterns which resulted in some original discoveries. INTRODUCTION
Mathematics for Liberal Arts (MLA) students will often claim that they hate math-ematics and that they are not good at it. Inquiry-based learning (IBL) providesa perfect opportunity for students to change that narrative as it allows them touse creativity and their own strengths to solve problems. I encountered such asituation in my MLA classes recently when my students were exploring the
RascalTriangle [1]. However, in order for my students to be in a position to be that suc-cessful, a classroom environment that supported inquiry needed to be established;this section describes the course and several factors that helped set the stage formy students discoveries.1.1.
Mathematical Explorations.
At Westfield State,
Mathematical Explorations ,our most popular MLA course, is populated by students who are typically from thehumanities or social sciences and have very negative attitudes toward mathematics[9]. One of the main goals in this course courses is to have our students experiencemathematics as an artistic, creative and humanistic endeavor. As a result, thetopics in the course are primarily from pure mathematics, (although the specificchoices are left up to the instructor), and the course is taught using a cooperative,inquiry-based learning approach. See [3] and [6] for more details about our class-room and [4] for a series of freely-available inquiry-based books we have written forthis type of course.The room in which most of our Mathematical Explorations classes are taught areequipped with nine tables at which the students work cooperatively in groups of 3to 5, while the instructor moves about the room answering and asking questions,offering encouragement and occasionally providing suggestions.
Key words and phrases.
Inquiry, Rascal Triangle, Mathematics for Liberal Arts, StudentPatterns.The author’s work as part of the
Discovering the Art of Mathematics Project ( http://artofmathematics.org ), which provides the basis for the curriculum in this course wassupported by NSF grants DUE-0836943, DUE-1225915, and a generous gift from Mr. Harry Lucas. During class time, the students are engaged in mathematical tasks, either workingthrough a set of carefully constructed questions designed to guide the students to(re)discover the solution, or working on an open-ended big question. It was thislatter approach that led to my students’ discoveries.1.2.
Course Aspects.Sharing:
Throughout the semester, students were asked to share ideas andpatterns with the class. This started with the a + 5 b problem: Whatare all the possible values for a + 5 b when a and b are non-negative inte-gers? See [5] for more details about solutions to this problem. The solutionsto this problem were not original, but students were asked throughout theinvestigation to share patterns they had found and why they believed thepatterns were true. This helped establish a culture of sharing ideas and thereasoning behind them, whether right or wrong, so by the time we got tothe Rascal Triangle exploration, the sharing of ideas was expected. As onestudent wrote, "Yet once we find a solution, something that actually worksfor the problem, it is exhilarating. We become so excited to explain this tothe teacher and present it to the class." The Proof : We watched the PBS special, "The Proof" [12] in class and thestudents were asked to read a portion of the Introduction to
Modular ellipticcurves and Fermat’s Last Theorem by Andrew Wiles [14] and identify verbsthat corresponded to actions taken by Wiles while working on the Taniyama-Shimura conjecture. The identification of these actions helped the studentssee that while the problems they work on are much simpler than Taniyama-Shimura, their process was similar to that undertaken by Wiles. They alsosaw that mistakes could be valuable and not something to be feared.
Mixing the groups:
I would change the groups regularly so that, as as onestudent put it, ". . . as classes went on, the entire class became friends, whichmade the experience [in this course] so much more enjoyable." Another stu-dent wrote:One other thing that was great about the environment of the classwas that I actually got to know my classmates. In some other strictlylecture based classes, I would often go a whole semester without evengetting to know probably around 80% of the class because there wereno opportunities to do so. In the group work environment however,it allowed us to bond with each other because we were all workingtowards a common goal, and I definitely was friendly with a strongmajority of the students in class by the end of the semester, and alsomade a few close friends.
Clapping Rhythms and Pascal’s Triangle:
Prior to exploring the RascalTriangle, I had my students do a clapping and rhythm activity from
Dis-covering the Art of Mathematics: Music [13, Chapter 2] that resulted in anexploration of patterns in Pascal’s Triangle. One student wrote,For instance, we found that when choosing a selected amount of num-bers diagonally, the sum of those numbers will allow the creation ofa "hockey stick" shape, as the sum is also a number within Pascal’striangle, making the formation possible. This activity was not aboutlong, complicated algorithmic equations, it was an act of discovery,observation and can even be seen as art.
TUDENT INQUIRY AND THE RASCAL TRIANGLE 3
These factors, and, of course, the students themselves, created an environment thatled to the discoveries described in Section .2. THE RASCAL TRIANGLE
This section contains a brief review of the Rascal Triangle if the reader is unfamiliarwith it.In 2010, middle school students Alif Anggaro, Eddy Liu and Angus Tulloch wereasked to determine the next row of numbers in the following triangular array:
Figure 1.
A triangular array.Instead of providing the row from Pascal’s Triangle that the instructor expected,1 4 6 4 1they produced the row 1 4 5 4 1.They did this by using the rule that the outside numbers are 1s and the insidenumbers are determined by the diamond formula
South = East · West + 1
North where
North , South , East and
West form a diamond in the triangular array asin Figure . Figure 2.
North , South , East and
West entries in a triangular array.Continuing with this rule Anggaro, Liu and Tulloch created a number triangle theycalled the
Rascal Triangle [1].
Figure 3.
The Rascal Triangle.
PHILIP K. HOTCHKISS
Because the diamond formula involves division, their instructor challenged Anggaro,Liu and Tulloch to prove that it would always result in an integer. They did this byusing the fact that the diagonals in the Rascal Triangle formed arithmetic sequences;see [1] for details.In the Spring 2015 semester, students in a
Mathematical Explorations classes taughtby one of my colleagues, Julian Fleron, discovered that the Rascal Triangle can alsobe generated by the rule that the outside numbers are 1s and the inside numbersare determined by the formula
South = East + West − North + 1 . This formula also follows from the arithmetic progressions along the diagonals.See [7] for details about this discovery; which, as best as can be determined, wasunknown at the time. Thus, the Rascal Triangle has the property that for anydiamond containing 4 entries, the
South entry satisfies two equations:
South = East · West + 1
North [1]
South = East + West − North + 1 [2]The fact that both Equations and can be used to generate the Rascal Trianglewas intriguing; and, as part of an effort to better understand the Rascal Triangle,I had one of my Mathematical Explorations classes explore number triangles ingeneral and the Rascal Triangle in particular. What followed was remarkable;my students became incredibly engaged with the exploration and some of themproduced some original results, which are described next.3.
STUDENT PATTERNS AND NUMBER TRIANGLES
After the exploration of Pascal’s Triangle, I gave my MLA class the first six rowsof the Rascal Triangle and asked them to find a method for extending the trian-gle in a manner that was consistent with the first six rows. The hope was thatthey would identify the arithmetic sequences on the diagonals and eventually (re)-discover Equations and/or . The only instruction given was the reminder thatPascal’s Triangle could be generated by the rule the South = East + West , sostudents were free to find any rule that made sense to them. Students found thisto be liberating; as one student wrote in her final refection (as a letter to a futurestudent):. . . Professor Hotchkiss gave us certain things, such as a number tri-angle, to find patterns in, every table’s group would dive in and getso into what they were trying to figure out. We ended up figuring outcrazy patterns, formulas, and equations in all kinds of different math-ematical things that were presented to us in class. The best part was,Professor Hotchkiss never told us there was ever going to be a verycertain correct answer. He told us to work through it and discover.Discover. . . That’s a word I never thought I would here [sic] in math.I always thought of math as a very set way of doing things with onlyright or wrong answers. I always thought there was no discoveringnew things in math, you just learned what was already established inthe text book.
TUDENT INQUIRY AND THE RASCAL TRIANGLE 5 . . . We, as a class, literally found new mathematic patterns in anew number triangle that have never been discovered before. Howcool is that? Pretty cool, I know.Another student wrote. . . instead of being told how to "do" the number triangles, Hotchkissencouraged us to look for patterns in them. Not a specific pattern,ANY pattern. This was vital because it put the students in a positionof power and gave them a thirst for finding more patterns. Once wegot frustrated with finding something new or hopeless with a specific"mini-prompt" we would get a bit of assistance from Hotchkiss thatwould sort of "nudge us along" if you will, almost like a dad givinghis son a little push while learning how to ride a bike.3.1.
Student Patterns in the Rascal Triangle.
Following "discovery" of theRascal Triangle via the arithmetic sequences on the diagonals, I asked my studentsto find other rules or patterns that would generate the Rascal Triangle. Equa-tions and had not yet been introduced, and as before, very little instructionswere given so students had the freedom to derive any patterns that made sense tothem.The patterns that were presented to the class were described using an informalnotation involving directions relative to the the South entry, as shown in Figure ,that was based on Angarro, Liu and Tulloch’s diamond description of the entriesin the Rascal Triangle in [1]. South EastWest NorthNW NENN NNEWNW ENENNW NNN
Figure 4.
Location of EntriesThe following definition was also helpful.
Definition.
For any number triangle, the diagonals running from right to left arecalled the major diagonals while the diagonals running from left to right are calledthe minor diagonals. (a) major diago-nals. (b)
Minor diago-nals.
Due to time constraints, proofs of the student patterns were not presented in class.However, they are accessible to undergraduate mathematics majors, and I would
PHILIP K. HOTCHKISS expect that they would be accessible to most MLA students as well. Proofs of thesepatterns (in a more generalized setting) can be found in [11].Several groups provided methods for extending the number triangle, but two ofthese were of particular interest. The first pattern was called the
T-Meg Rule .1. T-Meg Rule: Tim and Meaghan observed that starting with row 4 of theRascal Triangle, we could determine
South from
North and the first twoentries,
North and North , on North ’s row. That is,
South = North + North + North . Figure 6.
T-Meg Rule.For example,
16 =
North + North + North = 9 + 1 + 6 Note that T-Meg’s Rule also holds for the second entry in a row as well.
North + North + North = 1 + 1 + 6 As Meaghan wrote about her role in finding the T-Meg Rule (Example below)The hockey still pattern in Pascal’s triangle is what helped me find theT-Meg pattern in Rascal’s triangle. I knew there was a similar hockeystick pattern in Pascal’s triangle which made me look for other waysaddition was used in Rascals Triangle. This is how I was able to findthe T-Meg pattern. That was a huge turning point for me because Ihad found a pattern in the triangle that no one else had previouslyfound. That definitely made me more confident in my mathematicalabilities.2. Ashley’s Rule: Ashley observed that we could determine South from
West , East , NW plus a "diagonal factor". The factor was associated with theminor diagonal containing the South entry, and was equal to − k , where k = 0 corresponds to the minor diagonal consisting of all 1s. Figure 7.
Ashley’s Rule
TUDENT INQUIRY AND THE RASCAL TRIANGLE 7
For example,
19 =
West + East − NW − diagonal factor = 16 + 13 − − While looking for methods of generating the Rascal Triangle, one student observedtwo patterns involving diamonds within the Rascal Triangle.3. John’s Diamond Patterns:a. John’s Odd Diamond Pattern: The average of the n numbers that formthe edge of a diamond in the Rascal Triangle with n + 1 numbers oneach side is the number in the middle of the diamond. For example, forthe diamond in Figure , Figure 8.
John’s Odd Diamond Pattern.
13 + 16 + 19 + 25 + 31 + 26 + 21 + 178 = 1688 = 21 = = 21 b. John’s Even Diamond Pattern: If you form a × -diamond in the RascalTriangle and a diamond with n numbers on each side that has the × -diamond in the center, then the average of the n − numbers along theedges of the outer diamond is equal to the average of the 4 numbers alongthe edge of the × -diamond. For example, in Figure Figure 9.
John’s Even Diamond pattern
PHILIP K. HOTCHKISS
19 + 22 + 29 + 254= 954 = 23 . . = 47520 = 23 . . John’s original formulation of this pattern was that the the sum of the n − numbers along the edge of the outer diamond was n − timesthe sum of the 4 numbers along the inner diamond.
19 + 22 + 29 + 25= 95;11 + 13 + 15 + 17 + 25 + 33 + 41 + 36 + 31 + 26 + 21 + 16= 285 = 3 · = 475 = 5 · . CONCLUSION
The results in this paper are the results of explorations by Mathematics for LiberalArts students looking for patterns in the Rascal Triangle. In particular, the dis-covery that the Rascal Triangle could be generated by a several different additionrules, illustrate the power of "non-expert" eyes seeing patterns that the expertscould not see because they already knew the patterns. One student wrote at theend of the semesterOne moment when I was inspired by math in this class was whenwe were discussing the Rascalls [sic] triangle. It amazed me howthe students in my class whom we all worked together were able tofind proofs on things that very high level mathematicians had beenlooking at for years. I had never once been in an environment in aMath classroom in which I felt like everyone there was willing to helpand support each other. We were not treated like cogs in a machinethat we had no control of. We were treated like actually able bodiedhuman beings and were able to look at things in ways that were notcontrolled by the teacher.Theses student investigations inspired me to examine the algebraic structure of theRascal Triangle more deeply and resulted in the exploration of
Generalized RascalTriangles , see [11] for details.
TUDENT INQUIRY AND THE RASCAL TRIANGLE 9
ACKNOWLEGEMENTS
The author would like to acknowledge his students who observed the patterns dis-cussed in this paper: Ashley Craig, John Coulombe, Timothy Schreiner, MeaghanSparks and Evan Wilson.
References [1] Anggoro, A., Liu, E., and Tulloch, A., The Rascal Triangle,
The College Mathematics Jour-nal , , No. 5, November 2010, pp. 393-395.[2] Blair, R., Kirkman, E.E., Maxwell, J.W., Statistical abstract undergraduate programs in themathematical sciences in the United States: 2015 CBMS survey, American MathematicalSociety, Providence, RI, 2018.[3] Ecke, V., Our Inquiry-Based Classroom, retrieved from .[4] Ecke, V., Fleron, J., Hotchkiss, P. and von Renesse, C, Books: Inquiry-based Learning Guides,retrieved from .[5] Fleron, J., a +5 b Proofs, retrieved from .[6] Fleron, J., Classroom Vignette retrieved from .[7] Fleron, J., Fresh Perspectives Bring Discoveries,
Math Horizons , , No. 3, February 2017,p. 15.[8] Online Encyclopedia of Integer Sequences retrieved from https://oeis.org/ [9] Hotchkiss, P., Audience: Learning about our MLA Students, retrieved from .[10] Hotchkiss, P., Movie "Proof": How our students view the process of mathematics, retrievedfrom .[11] Hotchkiss, P Generalized Rascal Triangles, preprint, 2019.[12] "The Proof", NOVA , WGBH, Boston, 1997. DVD[13] von Renesse, C. with Fleron, J. Hotchkiss, P. and Ecke, V.,
Discovering the Art of Mathe-matics: Music , , 2015.[14] Wiles, A., Modular, elliptic curves and Fermat’s Last Theorem Annals of Mathematics , ,No. 3, 1995, pp. 443-551 Department of Mathematics, Westfield State University, Westfield, MA 01085
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