Students Produce Original Approaches to the Generation of Pythagorean Triples

  David Dennis

University of Texas at El Paso

Introduction

            The setting of this research was an initial pilot version of a reformed mathematics class for pre-service K-8 teachers at the University of Texas at El Paso (UTEP).  This course, entitled "Properties of Real Numbers," is the last mathematics content course that is required for all students certifying as K-8 teachers regardless of specialization.  Traditionally this course focused on running through the algebraic structures of the integers, rationals and real number systems in a largely formal way with some attention to problem solving activities.  The one prerequisite course is a standard college course that moves quickly through traditional algebra (linear and quadratic), logarithms, and some probability, where pre-service teachers are mixed with students majoring in business and nursing.

            In an effort to rethink possible mathematical options for the these future teachers, an initial research experiment was conducted as part of the first phase of a National Science Foundation project known in El Paso as the Partnership for Excellence in Teacher Education (PETE), which is one of a number of NSF's preservice teacher collaborative around the nation.  This experiment was conducted with one section of 36 students in the Fall of 1995.  The Department of Mathematical Sciences at UTEP and the College Education agreed to allow for complete freedom during this pilot to experiment with new curriculum and forms of assessment. 

            El Paso is an industrial urban center in far west Texas situated on the Mexican border next to Ciuadad Jaurez, Mexico, the fourth largest city in Mexico.  El Paso is one the poorest cities in the U.S.A. and is approximately 70% Mexican-American.  It is isolated by many miles of barren desert in all directions.  85% of the students at UTEP are drawn from the city of El Paso and 80% of the teachers in the El Paso area public schools are graduates of UTEP.  The situation has been described as a "closed loop."  In the pilot class to be discussed, 25 of the 36 students were Mexican-Americans.    

The Nature the Classroom Learning Environment

            The learning environment of this experimental mathematics class was modeled largely on the theories of Jere Confrey which are based largely on a radical constructivist framework with elements of social Vygotskian theory (1994b).  Several pieces of the curriculum were taken directly from Confrey's work on student cognition of ratio and proportion (1994a).  One main goal was to create what Confrey calls a "balanced dialogue between grounded activity and systematic inquiry."  The other main goal was to bring student voice and perspective to the foreground.  These intentions were implemented through a series of weekly projects where the students engaged with challenging mathematical investigations that all stemmed from direct physical situations.

            Assessment in the class was based on written project reports that were turned in weekly.  These reports were returned with written comments and students then had the option of rewriting each report for a higher grade.  Other than the 15 week semester itself, there was no limit on the number of times that a report could be rewritten.  A student's grade was based on top grades eventually achieved.  Project reports were given one of three grades.

√-       Some engagement with the problem, but substantial questions remain.

√        A well reasoned explanation, but some questions remain.

√+     A complete and thorough explanation, no further questions. 

The three possible project grades are a modified version of the situation faced by mathematical researchers where progress usually begins with some special cases

(√-), then moves to an explanation of all but a few special cases (√), and ends with a through argument (√+).  The student's in this class were not held to the linguistic standards of formal mathematical proof, but rather to a standard of peer review by their classmates in the context of the questions and comments of the instructor.  A √+ was given when proof beyond a reasonable doubt was achieved in the social context of the class.  In most cases such arguments could be easily transformed into formal proofs by a professional mathematician operating in his/her own social setting.  Students who eventually achieved a √ on 80% of the projects got a grade of C or better.  Students who eventually achieved a √+ on 90% of the projects received an A.  Students were informed of this policy at the outset of the class.

            The mathematical content of the curriculum fell into three basic sections.  The first several projects contained a variety of situations where integer data was collected from counting situations and the students sought ways to make predictions (e.g. the tower of Hanoi).  Both recursive and explicit statements were acceptable as long as they were well connected to the situation.  Some study of the relations between recursion and explicit statements were discussed using student generated strategies.  The second set of projects focused on the relations between integers, ratios, and geometry.  Beginning with geoboard problems (i.e. lattice geometry) students investigated lengths, areas, scaling, similarity, and quadratic irrational ratios.  For example, they were asked to find a strategy for finding all possible rescalings of a figure on a geoboard of arbitrary size.  The final projects dealt with data and statistical distributions from games and the algebraic structure of symmetry transformations.   For example, the symmetry group of a square was generated from the question, "how many ways can you pick up a cardboard square and return it to the same place on the table?" 

            Class time was spent in the following ways: short introductions to new projects, group explorations, class presentations after the first round of written reports, and brief lectures on the historical and cultural background of concepts.  Symbolic notation was introduced only as needed to deal with concepts that arose in the group explorations.  Student presentations were used to give struggling students a chance to learn from their classmates and to push the presenters to further articulate their ideas and arguments, thus allowing everyone to improve on their rewritten reports. 

            Student voice and perspective became the focus of most of the class discussions as various strategies and predictions were tested in a wider problem solving context and in relation to the social setting of the students.  The content of the curriculum was controlled by the instructor (myself) only to the extent that I provided the tools and asked the initial questions.  The ultimate direction of further investigations was largely shaped by the questions that arose in the class.  Many of the initial projects were adapted directly from my studies of the history of mathematics although only minimal historical background was provided directly to the students (Dennis, in press).  Students who desired more cultural and historical background were encouraged to engage in outside research or to take a newly-restructured course in the history of mathematics.

The Pythagorean Project Emerges

            In order to see how this process worked, this paper will examine in detail how a particular set of questions and strategies developed in the class.  In particular, the work of two students on one project will be described in detail.  The focus project itself was not one of the curricular intentions at the outset, but was developed in the middle of the class in response to questions that arose on the first set of geoboard (lattice geometry) explorations.  Initially students were asked to find all possible different sized squares that can be made with one rubber band on a geoboard[1] along with their areas.  After constructing squares of area 1, 4, 9, and 16 square units they found others of area 2, 5, 8, and 10 square units, that sit at odd angles with respect to the lattice.  Eventually most students showed that any line between any two points on a lattice can be used as a side of a square with all four corners occuring as lattice points, although articulating an argument for why this was true was, at first, quite difficult for them. 

            They were then asked to find the lengths of the sides of these squares using the distance unit of the lattice.  Most students attempted to measure the sides by making a ruler in the appropriate lattice units. Very few students used any form of square root calculation in the form of either the distance formula or the Pythagorean theorem, although this had been taught to all of them in previous formal mathematics classes.  One student was angry that I would even indirectly imply that he should have done this.  He said, "how can you expect us to use such an idea here when all we have ever done is memorize formulas to pass tests; the minute the test is over, we brain dump that stuff.  We've never used any of it except on math tests." 

            In response to the situation we discussed various forms of measurement and estimation.  I showed them an ancient Hindu method for finding integer fractions that approach the value of any square root.  Although difficult for them at first, this method has strong ties to basic geometrical ideas in that it produces a series of trimmings which construct a square of any given area (integer areas in our case).  They all mastered this technique and many were intrigued that they could now find a fractional approximation to a square root that was far more accurate than the 9 place decimal given by the square root button on their calculators.

            As we continued our investigations of scalings and ratios on the lattice using graph paper as an enlarged geoboard, some students noticed that not all diagonal lengths required a tedious square root calculation.  One student said, "hey sometimes you get lucky."  She had found that the diagonal of a 3 by 4 rectangle was exactly 5 units, since the square built on this segment had an area of exactly 25 square units (by dissection).  The most famous Pythagorean triple had been found and soon other multiples of it were discovered (like 6, 8, 10).  Several students thought that this was a real freak of nature and assumed that (3,4,5) and its multiples were unusual loners.  Unable to let this important historical moment pass by, I made the goal of the next class project the finding all such whole number diagonal lattice lengths under 100 (i.e. all Pythagorean triples or integers (a,b,c) such that a2 + b2 = c2).  I was well aware of the long cultural and historical role that this problem has played in mathematics for over four millennia.  Many students thought this was a silly project since all they had to do was take integer multiples of (3,4,5) up to 100.  The counter-example (5,12,13) soon emerged from early random searching.

            In the next two sections of this paper I will describe how two students searched for Pythagorean triples.  The two strategies employed were very different from each other and also quite different from anything that I have ever seen in my studies of the history of mathematics.  These student investigations are individually fascinating from the standpoint of educational and cognitive research.  They are also interesting in the way that each affected the direction of the other and of the rest of the class.  Even more surprising is that both of them contain original mathematical ideas which seem to be absent from any existing mathematical literature.  For this last reason I am violating the usual educational practice of using pseudonyms for subjects in educational research.  The mathematical ideas are attributed to the students using their full actual names. 

            Both of these students entered the class with hostile and negative feelings about mathematics and both were initially glad that this was their last required course in mathematics before certifying as K-8 teachers.  As we shall see, the nature of the class and their individual engagements brought about a variety of attitudinal and intellectual changes.  Although the two detailed stories to be presented contain some unique mathematical ideas, the changes in attitude undergone by these students were not unique.  A majority of students in the class underwent similar changes.  Besides the written documentation of the projects, videotaped interviews were conducted with 15 different students.  I shall return to these issues later.

Darron Saunders' Geometric Approach to Pythagorean Triples

            In both high school and at the University Darron Saunders had struggled with mathematics and often his struggle had been very frustrating.  Even when he did pass his classes he often came away feeling that he had learned nothing of any lasting value.  Formal symbols manipulated in meaningless ways is how he summarized most of his secondary and University mathematics, and hence he avoided all but the minimum mathematics requirements.  Saunders also has a long history of mild dyslexia which made all academic achievement more challenging for him.  As a pre-service K-8 teacher he concentrated on special education because he has natural affinities for students with learning disabilities.

            Like many dyslexics, Saunders has strong spatial and visualization abilities.  These abilities are rarely rewarded in an algebra-dominated mathematics classroom.  Saunders found outlets for his abilities mostly in the building trades and in sports.  Projects that involved shapes, geometric ratios and scaling came easily to him and he particularly enjoyed working with a geoboard, graph paper, ruler, and compass. 

            When the search for Pythagorean triples emerged from the class's investigations of lattice geometry, Saunders was loathe to abandon physical geometry for number theory.  While all other students in the class took up calculators, combined with tables and algebraic explorations, Saunders continued to search intensely for Pythagorean triples within the arena where they had originally emerged, lattice geometry.  Saunders drew many pages of figures on graph paper using his ruler and compass.  These figures were incomprehensible to other students in his group who ignored his investigation and went on with their numerical searches.  I looked at more than 15 pages of Saunders' first drawings and I could not see where he was going.  I gently remarked to him, "I think you're going to have to go numeric on this one."  He ignored my remark and I did not pursue it remembering Maria Montessori's principle, "never interrupt a child during a period of absorption."  Saunders is not a child but the principle seemed apt.

            Eventually Saunders began giving new Pythagorean triples to students who were attempting to hunt them down with calculator searches.  Saunders gave

(18,15,17), and (20,21,29) to some students who thought that the all Pythagorean triples were integer multiples of either (3,4,5) or (5,12,13).  Saunders' first written report contained many figures and few words of explanation and I could not at first understand what he was doing partly because as a historian of mathematics I was under the false impression that I knew all of the fundamentally different approaches to this problem.  After a discussion with Saunders his method of generation became clear.

            Using graph paper and a compass, he drew circles of integer diameter so that both ends of a diameter fall on lattice points in the same horizontal row.  The center of the circle may or may not fall on a lattice point depending on whether the diameter is even of odd.  He drew such a figure for each integer diameter separately.  Next he examined each circle to see whether it hit any other lattice points.  If it did, by symmetry, it hit four such lattice points, one in each quadrant.  Connecting any three of these four symmetrical lattice points yielded a right triangle where all three sides have integer lengths.  See Figures 1 and 2.  This method depends on knowing that any right triangle inscribed in a circle will have a diameter as its hypotenuse.  This geometrical concept is well known by carpenters who know how to use a metal square to draw a circle given the two endpoints of a diameter (i.e. pound in two nails at these endpoints and place the two legs of the square against the two nails, then slide the square with a pencil at the right angle). 

            Saunders' figures were an attempt to check for lattice points on each circle with an integer diameter under 100.  He used different scales of graph paper and he began taping sheets of paper together to get larger sheets.  He was well aware that care and accuracy were important here and his pencil-drawn figures were more clear and precise than the level of resolution of the computer-generated figures below.  

            Saunders was well aware that no matter how carefully he drew his figures some physical observational error might creep in especially as the diameters got larger.  For this reason when he found a possible lattice point on one of the circles he would then check the resultant Pythagorean triple by calculation to make sure that the sum of the squares of the legs equaled the square on the hypotenuse.  Occasionally he would have to discard a pseudo-Pythagorean triple where a circle seemed to hit a lattice point but the calculation showed that it was close but not quite there.  See Figure 3 for an example of what seems to be another set of lattice points on the circle of diameter 29, but where calculation shows that it must be a near miss.

            The important feature of Saunders' method is that, although some pseudo- Pythagorean triples may turn up and have to be discarded, the method does not miss any true Pythagorean triples.  It does systematically find all Pythagorean triples, but how can one be sure of this?  This became the question that I posed for Saunders and other students who came to understand and appreciate his method.  One thing that Saunders noticed is that all of the vertical legs (AB) of the triangles in his figures must have even integer lengths since they begin and end on lattice points and the horizontal diameter always bisects this leg at a lattice point.  Saunders wondered about whether every Pythagorean triple must have at least one even leg.  All of the examples found by the class always did.  When I pressed him on this point he reasoned that even if a Pythagorean triple existed where both legs were odd, its double would have even integer legs and would eventually show up using his construction method.  Therefore one must check each new Pythagorean triple for a possible common factor of two to make sure that the method generates all of them.  Of course this never happened since it is true that all Pythagorean triples do have at least one even leg.  Saunders was not interested in using his method to prove this number theoretic fact but only in arguing that his method is a fully general way to generate all Pythagorean triples.[2] 

            This again raises the issue of when a Pythagorean triple is primitive (i.e., the three numbers have no common factor).  This was an important issue for those who were hunting for Pythagorean triples numerically.  They knew that once they found a primitive one they could easily generate all of its multiples (similar figures to Saunders).  As we shall see in the next section, this issue became crucial for another student who approached the problem through the use of table recursions.        

Susanna Hernandez's Tabular Approach to Pythagorean Triples

            Susanna Hernandez came to the class with many insecurities about her abilities to engage in mathematics.  She said in an interview that she had often felt "dumb in math," although she did not generally consider herself a "dumb person" (see later section).  Like most citizens of El Paso, Spanish is her first language but her English was generally better than many other students at UTEP.  She plans to certify as a K-8 teacher specializing in bilingual education, and she hopes to teach at the second grade level.  As this mathematics course developed she gradually became more and more enthusiastic.  She found that the mathematics projects in the class were exciting to share with her extended Mexican-American family and this connection was influential on her view of the value of these projects.  The study group in which she participated both in and out of class worked in Spanish and her increasingly influential position in this peer group was an important factor in her change in attitude about her own mathematical abilities.  She said later that the Pythagorean triples project was the most important event in this evolution.

            The first few projects in the class involved discrete counting projects where looking at patterns in the differences and ratios in consecutive terms in a sequence had proved a fruitful strategy for many students.  These projects, however, had only involved making predictions in a single sequence of integers.  Hernandez found these early projects much more to her liking than the latter more geometrical ones.  She had developed a keen ability to find and understand recursive patterns, although her formal algebraic skills were poor.  At the point in the class when the Pythagorean triples project emerged she gladly returned to a systematic hunt for integer patterns using recursive techniques, but this time she faced a triple sequence and had to devise a much more subtle recursive technique.

            Several members of the class began this project by writing down a list of the first hundred perfect square numbers and then simply searching for any two that might add up to another member of the list.  A number of Pythagorean triples were found this way, but Hernandez and her group soon realized that many of these were multiples of each other and that by finding a primitive triple they could then multiply it by integers and quickly generate its "family group."  Thus for them the main problem became finding primitive Pythagorean triples. 

            The first three primitive Pythagorean triples that Hernandez's group found were (3,4,5), (5,12,13), and (7,24,25).  Hernandez placed them in rows and noticed that the progression of the smallest numbers, 3, 5, 7,  suggested an obvious arithmetic pattern.  She also noticed that the largest numbers (the hypotenuses) were each one more than the corresponding largest legs.  She next noticed that the largest legs increased by 12–4 = 8, and 24–12 = 12 = 8+4 , so she tried continue this pattern and generated Table 1, where the first column has a constant difference and the second column has a constant second difference and the third column is obtained by adding 1 to each entry in the second column.  Continuing the pattern she generated triples and was pleased to find that they all satisfied the Pythagorean relation and that they were all primitive.  Each new entry in the table could then be multiplied by any integer to generate a series of new families of Pythagorean triples.        

Table 1 - Pythagorean Triples where  c = b+1

            Hernandez and her group thought that perhaps they might have found all of the Pythagorean triples since they now had an infinite recursive list of primitive triples.  This hope was dashed by their classmates with the example of (8,15,17) which is primitive and does not appear in Table 1.  With help of her fellow group member, Sonia Manzano, Hernandez went to work on another table.  She thought that since her first table involved primitive Pythagorean triples using all of the odd numbers as legs (column a), perhaps the problem could be completed by generating a companion table where the first column consisted of the even numbers.  She rearranged (3,4,5) into (4,3,5) and then doubled (3,4,5) to get (6,8,10).  Putting these together with (8,15,17) she looked again for some pattern of recursive generation. 

            Hernandez found such a pattern as follows (See Table 2).  She first noticed that the hypotenuses (c) were always 2 more than the largest legs (b).  She then noticed that by adding b+c in the first row it yielded the value of b in the second row (3+5 = 8).  Although this does not work for moving from the second row to the third row it can be fixed up by subtracting the previous value of b, i.e. b+c–(previous b), or in Table 2,  8+10-3 =15,  15+17-8 = 24,  24+26-15 = 35, . . .   Stated in the algebraic language of recursion she generated Table 2 by using the relations:  bn+1 = bn+cn–bn-1, an+1 = an +2, and cn = bn+2, although Hernandez herself never used such algebraic language.  After checking that this method did indeed generate a table of Pythagorean triples, she later realized that she could also think of Table 2 in the same way that she had generated Table 1 where the first column has a constant first difference, and the second column has a constant second difference (i.e., the values of b go, up 5, up 7, up 9, etc.)

            Hernandez noticed that only every other one of the rows in Table 2 is a primitive Pythagorean triple.  This was not terribly surprising since she had used a double of (3,4,5) to get the recursion going in the first place.  In fact the double of every row in Table 1 appears in Table 2 interspersed with an infinite list of new primitive Pythagorean triples.  Hernandez now felt convinced that she could generate all Pythagorean triples by taking integer multiples of the primitives that occur in her two tables.  After all she had now covered all possible cases of values of a, both odd and even, and no one in her group had any counter-examples to her theory.  This was what I read in her first written report on this project.

Table 2 - Pythagorean Triples where  c = b+2

            When I read Hernandez's report I was quite amazed in several ways.  The Pythagorean triples in Table 1 are a special subfamily of all primitive Pythagorean triples that have a long history although I had never seen this tabular method used to generate them.  This table made sense in many ways and helped me to rethink some issues in the history of ancient mathematics where all too often modern historians speculate on ancient methods using algebraic language that can obscure the original concepts.  Table 2 was more surprising to me because I had never before seen any numerical or algebraic method that generated that particular list. 

            When this first round of reports was returned with comments, Hernandez presented her results and methods to the entire class.  Most students were intrigued by her patterns of generation.  Saunders and several others presented their work which included the example of (20, 21, 29) which is a primitive Pythagorean triple that does appear in either of Hernandez's two tables.  Saunders was particularly impressed with her tables because of the ease with which with one can generate new triples and because they were so alien to his way of thinking.  I added one comment of my own to the discussion which was that Table 1 could have started with the flat, degenerate "triangle" (1,0,1) and likewise Table 2 could have started with (2,0,2).  I pointed out that these flat triangles fit perfectly well into the recursion patterns that Hernandez was using.  Saunders pointed out that even if we considered these flat triples as degenerate triangles they still did not fit with the original search for integer diagonals on the geoboard, since these did not involve any real diagonals.  It was only during this class discussion that I came to fully understand Saunders' method of constructing triples.

            While waiting for the next class meeting I could not resist doing my own round of intensive research since I had just been shown by my students two methods for generating Pythagorean triples which I had never seen before.  A search through standard works in number theory and history of mathematics revealed no mention of either Saunders' or Hernandez's methods.  Oddly enough the only connection that made any real sense was between Hernandez's tables and the geometric construction of Pythagorean triples given by Euclid in Book 10, Lemma 1 before Proposition 29.  Euclid's construction can be viewed as using the parameter of a constant difference between one leg and the hypotenuse.

            When the class met again we continued our full class discussion and began constructing Table 3.  Observing that (20,21,29) had a difference of 8 between the large leg and the hypotenuse, we looked for others with that difference.  Following Hernandez's method we found others by either multiplying previous triples or by rearranging them.  (5,12,13) could be written as (12,5,13) and any triple in Table 2 could be multiplied by 4.  This was enough to get us started on a pattern which led to Table 3. 

Table 3 - Pythagorean Triples where  c = b+8

            Saunders' continuing ruler and compass search had led him to find (33,56,65).  This led the class to construct Table 4 where the hypotenuses are all 9 more than the second legs.  Again the pattern was started by multiplying rows in Table 1 by 9 and by rearranging other primitive triples like (8,15,17) and (20,21,29) into (15,8,17) and (21,20,29).  In this table we were all surprised to find that instead of finding an alternating pattern of primitives and multiples we found that there were two primitive triples between multiples from Table 1.  After the first two reversed primitive rows, all of the others were new primitive triples.

Table 4 - Pythagorean Triples where  c = b+9