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Alyssa Farrell Recursive Patterns and Polynomials Alyssa Farrell Iowa State University MSM Creative Component Summer 2006 Heather Thompson, Major Professor Irvin Hentzel, Major Professor Patricia Leigh, Committee Member Farrell 2 Patterns have been studied for centuries. Some of these patterns have been graced with names of the people who are most remembered for their work with them or related topics. Among these are the Fibonacci numbers, Lucas numbers, Pascal’s Triangle, Chebyshev’s Polynomials, and Cardan Polynomials. Patterns are used as building blocks in education from newborn babies and young children all the way through high school and beyond into post-secondary work. We were introduced as early as infancy to learning to recognize patterns, meaning both similarities and differences, when learning colors, shapes, the alphabet, and numbers. As we grew, we were introduced to new mathematical patterns like learning to count by 2’s, or 5’s. As we were more formally introduced to shapes, we learned to recognize that a square has four sides the same length, but unlike a kite, the square has four angles that are 90 degrees. Then we learned to add, subtract, multiply, and divide. All of these necessary operations were taught using a pattern-approach. Do you remember learning to read? We were taught to recognize patterns of letters which made certain sounds and that when certain sounds were strung together they made words. These patterns continued to be used as we developed higher in our math education. Teachers in junior high or middle school introduced us to an abstract idea of letting letters represent numbers as variables. We found out that adding “letters” was similar to adding numbers. In high school we elaborated on algebraic thought processes. Learning to multiply two binomials could be described using a pattern. Mathematics itself has been described as the “science of patterns” (Smith, 137). This description places a direct emphasis on the ability of math student to recognize, process, and discuss patterns. Farrell 3 Recursive thinking, or thinking with respect to patterns, has been a topic of many documents. The National Council of Teachers of Mathematics recommends in their 2000 publication, Principles and Standards for School Mathematics, that patterns are introduced as an integral part of math curriculum as early as preschool. They recommend that students continue being exposed to patterns through college. The importance of understanding patterns is addressed as an integral piece of knowledge for students to be able to use the process standards: problem solving, reasoning and proof, communication, connections, and representation. Patterns are an important part of mathematics curricula and also have more impact on a student’s life than just in their mathematics career. “…mathematical habits of mind are the most important things students can take away from their mathematics education…the real utility of mathematics is that it provides you with the intellectual schemata necessary to make sense of a world in which the products of mathematical thinking are increasingly pervasive in almost every walk of life… Recursive thinking also gives students genuine intellectual power.” (Cuoco). Patterns continue to be building blocks in learning new math. Even as I continue my education with my master’s degree, I use patterns to help me describe what happened; learning to relate patterns to new material made this learning easier. This is why I chose to look deeper into Thomas Osler’s article “Variations on a Theme from Pascal’s Triangle”. At first, I was drawn to the patterns and generating functions of any variation of Pascal’s Triangle, but as I delved into these, I realized there was more out there to explore than just the patterns of Pascal’s Triangle. Farrell 4 I started by looking at patterns developed when binomials were expanded and 2 i multiplied by a trinomial (a2 x + a1x + a0 )(b1x + b0 ) . This was interesting to me because of the ties it had within Algebra I and Algebra II curriculum. What I found was that there was a formula that could be used to find the coefficients of the polynomial for any value of i . i−c+2 c−2 i−c+1 c−1 i−c A()i,c = C(i,c − 2)⋅b1 ⋅b0 ⋅ a0 + C(i,c −1)⋅b1 ⋅b0 ⋅ a1 + C(i,c)⋅b1 ⋅b0 ⋅ a2 , where A(i,c) refers to the coefficient of the cth term in the polynomial expansion of 2 i (a2 x + a1 x + a0 )()b1 x + b0 and C(n,r)represents the combination of n things taken r at a time. This follows from the binomial expansion, which states that given a binomial, i (b1 + b0 )can be expanded using the formula: i i i−1 i−2 2 i ()b1 + b0 = b1 + C(i,1)⋅b1 ⋅b0 + C(i,2)⋅b1 ⋅b0 + ...+ b0 . To find a specific term, say i−c+1 c−1 the cth term, one can use the formula: C(i,c −1)⋅b1 ⋅b0 After working with the above idea and then moving on to work with the expansion of trinomials, I realized that this work had already been done. I went back to Osler’s article again and looked at other patterns he had unmasked during his explorations. He writes about the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, and also the Cardan polynomials and their relation to the Lucas polynomials (216-23). Farrell 5 Fibonacci Numbers One variation that Osler looks at is the following (218): 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 This variation of Pascal’s Triangle is formed by the expansion of the polynomial xm (1+ x)m . If one adds the columns of the arrangement, the sums are 1, 1, 2, 3, 5, 8, 13… This series is the well known Fibonacci numbers. The Fibonacci numbers are a series of naturally occurring numbers. The most familiar place where the Fibonacci numbers are represented is in the solution to the “rabbit problem” and in the family tree of a honeybee (Anderson, Frazier, and Popendorf). Fibonacci numbers also have a very close relation to the Golden Ratio (Knott, Fibonacci Number). Leonardo Pisano, better known by his nickname “Fibonacci”, meaning son of Bonacci, (1170-1250) is credited with solving the infamous rabbit problem, whose solution represents the series of numbers known by his name(O’Connor and Robertson, Fibonacci). The rabbit problem is stated as “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” (Anderson, Frazier, and Popendorf). The solution is represented by the table below, where M represents the number of the month and P represents the number of rabbit pairs in the population at the end of that month. Farrell 6 M 0 1 2 3 4 5 6 7 8 9 10 11 12 P 1 1 2 3 5 8 13 21 34 55 89 144 233 So, at the end of the 12th month, there are 233 pairs of rabbits in the population. This, however, is a very unrealistic problem, since we know that every birth does not necessarily produce a pair, that the population would have to be reproducing within itself and mutations are bound to occur, and that the food supply is unlimited. What are the chances that no deaths occur in this population for an entire year? Another example where the Fibonacci numbers are displayed is in the family tree of a honeybee. Since honeybees have a strange parentage, their family trees are very interesting. Some bees have two parents, some have only one. In the honeybee colony there are three types of bees: drones (males), workers (females) and a queen (special female). A queen is the only female that reproduces and her offspring can be male, produced from a non-fertilized egg, or female, from a fertilized egg. Thus a male honeybee has only one parent, the queen bee, while a female honeybee has two parents, both a male and a female. If we look at the family tree of a male honeybee (1 bee in generation 1), we see that he has only one parent, a female (1 bee in generation 2). This female honeybee has two parents, a male and female (2 bees in generation 3). Of the bees in generation 3, the male has one parent, a female, the female has two parents, male and female (3 bees in generation 4). The diagram and table below shows the number of bees in each generation. Farrell 7 M Generation 1 F Generation 2 M F Generation 3 F M F Generation 4 Generation 5 M F F M F Generation 1 234567 8 9 1011 12 Number of bees 1 123581321345589 144 This series of numbers, Fn ,can be defined as the recursive function Fn = Fn−1 + Fn−2 with F1 = 1and F2 = 1 (Knott, General Fibonacci). Any specific Fibonacci number, Fn , can be found by using the closed formula n n 1 ⎡⎛1+ 5 ⎞ ⎛1− 5 ⎞ ⎤ F = ⎢⎜ ⎟ − ⎜ ⎟ ⎥ . Simplified, this is the same as Binet’s Fibonacci n 5 ⎢⎜ 2 ⎟ ⎜ 2 ⎟ ⎥ ⎣⎝ ⎠ ⎝ ⎠ ⎦ n n 1+ 5 − 1− 5 Number Formula F = ()( ) (Knott, Fibonacci Number). n 2n 5 It can easily be proven that Binet’s formula satisfies the recursion formula with a proof by induction. To start with, we need to show that Binet’s formula works for n =1, n = 2 and n = 3.
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