2 class periods (1 period to introduce number cycles and the problems, another to discuss solutions)
- Understanding of recursively defined sequences
- Fluency with operations on rational expressions
- Knowledge of quadratic formula and/or equation solving by graphing
Ohio Standards Alignment
- Graphing calculators (helpful but not necessary)
- Spreadsheet software (helpful but not necessary)
- Printouts of empty number cycles are helpful (though easily created by students by hand)
Recursion, algebraic proof, simplifying rational expressions, solving quadratic equations
In this investigation, students investigate number cycles, that is, repeating number patterns. For instance, the sequence 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5, 7 . . . is a 4-number cycle since the four terms 1, 3, 5, 7 repeat. Initially, students are provided with a recursive formula that generates a 5-number cycle. They choose values for the first two terms of the sequence, then use the recursive formula to discover the cycling pattern. The cyclic nature of the recursive formula is surprising to many students. Since the formula appears to cycle regardless of starting values, students are naturally motivated to determine whether or not the formula always works. This leads to some interesting and unexpected connections to the golden ratio.
- To successfully generate terms in a sequence using a recursive definition.
- To generate hypotheses regarding the cyclic nature of certain sequences and to prove these hypotheses using algebraic arguments.
- To solve quadratic equations using the quadratic formula.
After students have several minutes to generate terms of their sequence using starting values of their own selection (or those provided by the teacher), they quickly notice that the terms repeat. This raises a series of questions regarding the sequence:
Is it possible to select starting values so that the first five terms are all negative values?
Is it possible to select starting values so that the first five terms are all the same value?
Investigate these questions, as well as several that you generate on your own. In each case, support your answer with mathematical work that confirms your contentions.
Does the sequence ALWAYS repeat regardless of the starting values one picks? Can you find starting values that "mess up" the sequence so that it doesn't repeat?
Present an empty 5-cycle diagram (see Figure 1) on an overhead transparency or whiteboard in a whole-class setting. Explain to the students that they will be placing numbers in each of the 5 empty spaces (i.e., circles) of the diagram.
Ask students to provide two non-zero values to put in the first and second spaces of the cycle. Initially, whole number values are preferable (until the students feel comfortable with the procedure for determining values in subsequent spaces). The values in subsequent spaces are calculated recursively using the following formula:
number in nth space = ((number in n-1 space)+1)/(number in n-2 space)
Based on the level of the students, you may wish to describe the formula without referring to the "nth term." For instance:
Next = (Now + 1)/(Previous)
Model the calculation of numbers in the third through fifth spaces in a whole group setting. Figure 2 illustrates the values of the five spaces in the cycle when the first and second spaces are set to 1 and 2, respectively.
Once students have successfully calculated values for spaces 3, 4, and 5, calculate the value that would appear in the next (i.e. sixth) space. In the above example, these calculations appear similar to the following:
Next = (Now + 1)/Previous = (1 + 1)/2 = 1
You may wish to remark that the value in the sixth space actually matches the value that students chose for the first space. "Hmm! That's strange! The values match! I don't think I've ever seen that one before. Let's try this with different starting values and see what happens."
At this point, students may be placed in small groups (e.g., pairs) and asked to repeat similar steps for different starting values of their own choosing. Students may wish to use calculators; however, if you wish to provide practice with fraction arithmetic, paper and pencil work may be preferable.
After 10 minutes or so, ask various student groups to share their values for spaces in the cycle. It will soon become apparent to students that the values of the sequence appear to "cycle" (i.e., the value in the first space always equals the value in the sixth space, regardless of starting values).
At this point, one may wish to explore yet more examples in a whole-class setting with a spreadsheet (see attached spreadsheet file) or Computer Algebra System. For any non-zero starting values, the sequence will repeat.
It is important to emphasize to students that a handful of examples provides evidence that the sequence may, in fact, cycle for arbitrary starting values in spaces 1 and 2. However, a handful of examples does not constitute a rigorous proof. The quest to prove that the pattern holds in general motivates an algebraic proof.
The following questions may be explored in class or offered as problems for homework:
Using algebra, do the following:
Prove that regardless of the values chosen for the first and second spaces, the number in the sixth space (i.e., the value derived from the fourth and fifth spaces) will equal the number in the first space (that is, prove that the numbers cycle in the diagram for any non-zero starting values).
Find initial "seed" values for the first and second spaces of the cycle that result in the same number appearing in all five positions in the cycle.
This activity provides students with opportunities to build and test conjectures while applying their knowledge of symbolic algebra in a novel situation. The task is easily differentiated for students of varying background. Changing initial values of the sequence significantly alters the complexity of subsequent student work. Some students may wish to explore the variable starting terms; others may explore integer, rational, and/or negative starting terms based on comfort and previous algebraic experience.
Arguably the two most critical aspects of this activity (if one wishes to teach it in the manner in which it was originally intended) are the following:
- First, spend some time modeling the calculations of an entire cycle with students, but also let them explore the 5-number cycle with a partner. Don't rush the activity in order to pose the two questions at the end. It will likely take students time to feel comfortable with the recursive definition that generates terms of the sequence. If students have previous experience with the Fibonacci sequence (particularly the recursive definition), the activity can build on this previous knowledge. In fact, this lesson serves as a nice follow-up to Fibonacci activities.
- Second, allow sufficient time for students to explore the problems posed at the end of this lesson. These two items require significant mathematical thought. I encourage you not to assign additional homework above and beyond the two problems posed in this activity.
The activity lends itself nicely to differentiation. In the same classroom, several students may be working on the general proof of the sequence, while others are testing various starting values in an effort to find a value that yields identical entries in all cells. The problems are well-suited for use with technology or paper-and-pencil procedures.
Extensions: Several interesting questions may be explored beyond those posed in this activity. For instance, students with significant experience with recursive formulas may wish to alter the recursive formula that generates the 5-cycle and analyze the results of various modifications. For instance, is it possible to modify the recursive formula to generate a 6-cycle, a 7-cycle, an n-cycle? Do all recursive formulas generate cycles? Interested students may wish to explore Mobius transforms (the recursive formula investigated in this activity is one example of a Mobius transformation). Students may also wish to write computer and/or CAS-based programs to investigate cyclic patterns both for specific numerical entries or in general (making use of the symbolic manipulation features of CAS).
The complete problem solution is attached.
Recursive Sequences / Fibonacci Numbers
Devlin, K. (2002, April). Numbers in the garden and geometry in the jungle. Mathematics Teaching in the Middle School, 7(8), 422-425.
This opinion piece discusses number patterns and Fibonacci numbers found in nature.
Miller, C. B., & Veenstra, T. B. (2002, January). Fibonacci: Beautiful patterns, beautiful mathematics. Mathematics Teaching in the Middle School, 7(5), 298-305.
Using examples from nature, the authors introduce the Fibonacci sequence and then describe some fascinating mathematical patterns, involving multiples and greatest common factors, that exist among these numbers.
Jarvis, D. (2007, April). Mathematics and visual arts: Exploring the golden ratio. Mathematics Teaching in the Middle School, 12(8), 467-471.
Mathematics and visual arts have long shared aspects of both form and function. This article introduces the reader to one such ancient connection - the golden section - presenting a historical, mathematical, and pedagogical context.
Extensions / Theory
Lyness, R. C. (1945). 1847. Cycles. The Mathematical Gazette, 29(287), 231-233.
Lyness, R. C. (1961). 2952. Cycles. The Mathematical Gazette, 45(353), 207-209.
These articles are primarily intended for teachers and include discussions of the upper level mathematics that underlies the cyclic nature of the 5-cycle.
Adapted by Michael Todd Edwards from Snover, S. L. (1982, March). Five-cycle number patterns. Arithmetic Teacher, 29(7), 22-26.