Virginia Mathematics Teacher Spring 2017

Recursive functions can also be used to spark discussions about numbers. For example, when first being shown the infinite continued fraction above, one student asked, “Is that even a number?” This led to a short discussion on the relationship between rational and irrational numbers. For more information on continued fractions, see Olds (1963) and Khinchin (1997). References Khinchin, A. Y. (1997). Continued fractions (3rd ed.). New York: Dover Publications. NCTM. (2000). In Carpenter J., Gorg S. (Eds.), Principles and standards for school mathematics . Reston, VA: The National Council of Teachers of Mathematics, Inc. Olds, C. D. (1963). Continued fractions (Vol. 18). New York: Random House. VDOE. (2009). Curriculum framework: Algebra II. Retrieved from http:// www.doe.virginia.gov/testing/sol/ frameworks/mathematics_framewks/2009/ framewk_algebra2.pdf

find that

=

, which is

approximately 1.302775. Discussion

This continued fraction task is a great introduction to recursive thinking. There are many entry points and possible extensions. As shown above, students can use their knowledge of rational numbers to generate patterns that will eventually lead to the ability to write recursive equations. Another pre-service teacher wrote, “I really had no clue how to start this, but I started with what seemed to be the simplest case and built it from there (see Fig. 4). I realized it’s actually quite simple: start with 3. Add 1 and make it a new 3 divided by that 1+3 we now have.” This young man went back to his knowledge of rational numbers in order to make sense of the continued fraction.

Nicole Fratrik Doctoral Fellow

University of Virginia Naf3pv@virginia.edu

Figure 4. First steps done by a pre-service teacher

Recursive thinking is advocated by NCTM and VDOE in the high school curriculum. The example shown above and other continued fractions can be a way to initially engage students in recursive thinking. The three strategies shown above for the infinite continued fraction lend themselves to Algebra II and PreCalculus. For Algebra I, one could also use finite continued fractions that converge to rational numbers (see Figure 5).

*A photos of the other author was not provided

Figure 5. Finite continued fractions

Virginia Mathematics Teacher vol. 43, no. 2

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